Have you looked at the tense and mode cho­sen for these vers­es in oth­er trans­la­tions that use lan­guages with a more clear­ly de­lin­eat­ed tense and mode in­flec­tion­al sys­tem than English has?

Here are two.

Latin

The Latin Vul­gate has for the first verse’s “The LORD is my shep­herd, I shall not want”:

Do­mi­nus re­git me, et ni­hil mi­hi de­er­it.

where de­er­it is the 3rd sin­gu­lar fu­ture ac­tive in­dica­tive of de­sum, mean­ing to be want­ing or lack­ing (to some­one), and then for the fourth verse’s “I will fear no evil”:

non ti­me­bo ma­la

where ti­me­bo is the first-per­son sin­gu­lar fu­ture ac­tive in­dica­tive in­flec­tion of ti­me­re, mean­ing to fear.

Spanish

The Span­ish Rei­na-Va­le­ra An­ti­gua with its KJV-equiv­a­lent old-timey lan­guage has for the first verse:

JE­HO­VA es mi pas­tor; na­da me fal­ta­rá.

where again fal­ta­rá is the third-per­son sin­gu­lar fu­ture in­dica­tive in­flec­tion of fal­tar, mean­ing to be lack­ing (to some­one), and for the fourth verse:

No te­me­ré mal al­gu­no

where again te­me­ré is the first-per­son sin­gu­lar fu­ture in­dica­tive in­flec­tion of te­mer, mean­ing to fear.

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Conclusion

So both the Latin and the Span­ish em­ploy the same sim­ple fu­ture in­dica­tives, with­out any signs of the sort of de­on­tic/epis­temic shad­ing that can in the­o­ry oc­cur with the English modals will and shall.

Be­cause those oth­er two trans­la­tions make no dis­tinc­tion there, I would ad­vise against read­ing too much in­to the KJV trans­la­tors’ par­tic­u­lar choic­es of will/shall in these two par­tic­u­lar vers­es.

In order to analyse a translation we have to look at the original it was translated from, although other translations of the same original may give additional insight. But that is where the problems begin:

The original is in Hebrew but I was not sure that was what the KJV was translated from so I looked it up. The answer is so complicated that I can only suggest you read the Wikipedia article, or, at the very least the sections on Considerations for a new version and Literary attributes. The most significant passage is

Old Testament

For their Old Testament, the translators used a text originating in the editions of the Hebrew Rabbinic Bible by Daniel Bomberg (1524/5),[132] but adjusted this to conform to the Greek LXX or Latin Vulgate in passages to which Christian tradition had attached a Christological interpretation.[133] For example, the Septuagint reading "They pierced my hands and my feet" was used in Psalm 22:16 (vs. the Masoretes' reading of the Hebrew "like lions my hands and feet"[134]). Otherwise, however, the Authorized Version is closer to the Hebrew tradition than any previous English translation – especially in making use of the rabbinic commentaries, such as Kimhi, in elucidating obscure passages in the Masoretic Text;[135] earlier versions had been more likely to adopt LXX or Vulgate readings in such places. Following the practice of the Geneva Bible, the books of 1 Esdras and 2 Esdras in the medieval Vulgate Old Testament were renamed 'Ezra' and 'Nehemiah'; 3 Esdras and 4 Esdras in the Apocrypha being renamed '1 Esdras' and '2 Esdras'.

The net result is that there was a range of considerations, but that the main one, at least in passages that were not politically or theologically contentious, was the Hebrew.

So we now turn to the Hebrew text

מִזְמֹ֥ור לְדָוִ֑ד יְהוָ֥ה רֹ֝עִ֗י לֹ֣א אֶחְסָֽר׃

גַּ֤ם כִּֽי־אֵלֵ֨ךְ בְּגֵ֪יא צַלְמָ֡וֶת לֹא־אִ֘ירָ֤א רָ֗ע כִּי־אַתָּ֥ה עִמָּדִ֑י שִׁבְטְךָ֥ וּ֝מִשְׁעַנְתֶּ֗ךָ הֵ֣מָּה יְנַֽחֲמֻֽנִי׃

(Sorry I haven't figured out how to right justify that - can anybody help!)

Now I have a problem because I can't read it, so I turn to

Interlinear v1 and

Interlinear v4

where I learn that the two relevant verbs

2637 [e] ’eḥ·sār. אֶחְסָֽר׃ I shall want V‑Qal‑Imperf‑1cs

3372 [e] ’î·rā אִ֘ירָ֤א I will fear V‑Qal‑Imperf‑1cs

are parsed identically. Although I do not fully understand what is going on, Hebrew tends not to mark tense (as in present, future, etc.) explicitly, but they are both marked as "Imperf" which I think means a continuous sense. But the net result is clear - there is nothing in the original to say that the tenses differ between the two clauses.

So if there is no semantic difference, what is the difference? There are two issues that might affect the translation beyond the semantics. Firstly, this translation, more than most others, aimed for English that sounded good. Secondly, this book in the Bible is meant to be poetry. Some translations (known as Metrical Psalms) aim specifically for a fixed metre, so they can be fitted to a tune. This translation is not one of those, but they would, nevertheless, have taken the natural rhythm of English into account so as provide a text that sounded good when read out in a church. English has a stress on approximately every second syllable and if I try to identify the metre in this passage I get:

{a PSALM of DAVID.} the LORD is my SHEPherd; i SHALL not WANT.

he MAKeth ME to lie DOWN in green PASTures: he LEADeth me beSIDE the STILL WATers.

he resTOReth my SOUL: he LEADeth me in the PATHS of RIGHTeousness for his NAME'S SAKE.

YEA, though i WALK through the VALLey of the SHADow of DEATH, i will FEAR no EVIL: for THOU art WITH me; thy ROD and thy STAFF they COMfort me.

So the first verb is stressed and the second is not. Of course you do not have to stress it this way, but it sounds best to me and so it could have done to the translators. I think SHALL sounds best in stressed position and will in unstressed position. In fact, based on my subjective experience of the KJV, I don't think WILL was ever stressed.

So, in conclusion, I think the difference was that it sounded best with the first verb stressed and not the second, and that they used SHALL when they wanted stress and will when they didn't.

I used to teach English, French, and Spanish. I too have wondered about the difference in these two verbs. I have noticed in my own reading of the Bible throughout the years that “will” seems to have to do with the desire while “shall” has to do with "matter-of-fact ‘this is going to happen’ .”

"Shall" can almost be dispensed with in modern English except for its use in questions, specifically in first-person singular and plural, e.g.

Shall we dance? (is ubiquitous)

Will we dance? (is possible in some dialects)

Shall I change the TV channel? (usual)
Note that this is a request for information. It means 'would you like me to change the channel?'

Will I change the TV channel? (Used in some regions. I believe mainly Ireland)
For most native speakers, the sarcastic answer to this would be, "I don't know. I can't predict what you are going to do!" For Irish speakers, the reply might be "Yes please."

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The history is complicated and I can do no better than recommend taking a look at the Wikipedia explanation of the subject.

An illustration of the supposed contrast between shall and will (when the prescriptive rule is adhered to) appeared in the 19th century,[11] and has been repeated in the 20th century[12] and in the 21st:[13]

I shall drown; no one will save me! (expresses the expectation of drowning, simple expression of future occurrence)

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I will drown; no one shall save me! (expresses suicidal intent: first-person will for desire, third-person shall for "command")

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Shall and will - From Wikipedia, the free encyclopedia

The rule that shall goes with I and will goes with you, he, she, it, and they is no longer used in current-day English, but it came after the King James Bible.

You can see that the rules changed between the 17th century and the 19th century by comparing the frequency of I will and I shall in different works. In Middlemarch, by George Eliot, I will is used 114 times and I shall is used 216 times, nearly twice as many. In the King James Bible, I will is used 2042 times and I shall is only used 196. For contrast, in Middlemarch, it will is used 52 times, and it shall is used once. However, in the KJV, it will is only used 19 times, while it shall is used 682 times.

I don't actually know the full rules that Shakespeare and the King James Bible used for shall and will, but one aspect is that shall was more used just for simple futurity, while will was for volition on the part of the subject. Since when you use I will, volition is usually involved, one would expect I will to be more common. And since things do not have volition, you would expect it shall to be more common.

I believe the OP's example reflects this:

The Lord is my shepherd; I shall not want.

Here, whether I want or not is beyond my control, so I use shall.

Yea, though I walk through the valley of the shadow of death, I will fear no evil.

Here, I have some control over whether I will fear evil or not, so I use will.

The actual rules were probably considerably more complicated than this (although maybe the people who translated the Bible broke them when they thought it sounded better); will is occasionally used with it for things that have no volition (As in Matthew 16:13 "And in the morning, It will be foul weather to day: for the sky is red and lowring.") And sometimes I shall and I will are used in ways that seem to contradict this rule.

The Wiki article referred to states

Historically, prescriptive grammar stated that, when expressing pure futurity (without any additional meaning such as desire or command), shall was to be used when the subject was in the first person, and will in other cases (e.g., "On Sunday, we shall go to church, and the preacher will read the Bible.")

The article continues to the effect that this distinction has now been all but lost

The distinction was justified by the argument that arises from the meaning of shall and will

Shall expresses a firm resolution as to certainty of a future action. As you cannot make a resolution for someone else, "shall" is only appropriate to the first person. Shall also expresses a nuance of making every effort to achieve the action.

"I shall marry James and you will not stop me!"

Will, on the other hand expresses, a predicted or desired state of future affairs. It carries less force than shall and accepts that the future cannot be foretold with accuracy.

Thus

The Lord is my shepherd; I shall not want.

Means The Lord is my shepherd; I am absolutely certain that the future holds no lack of what I require: I have resolved not to want

The Lord is my shepherd; I will not want.

Means The Lord is my shepherd; I am pretty confident the future holds no lack of what I require. I can't imagine myself wanting.

The Wiki article also mentions the legal use of shall

These experts recommend using shall but only to impose an obligation on a contractual party that is the subject of the sentence, i.e., to convey the meaning "hereby has a duty to."

This is not quite accurate: what is being required is that the a contractual party is being asked to make a resolution that is binding upon himself - thus the "first person" aspect is maintained.

e.g. "The seller shall ensure that the product is safe."

It will be noted that "should" is often also used - this use approaches "must" but falls short as the law recognises that there may be circumstances in which "ensuring" is not possible/practical.