Aw, shucks!

Your ear is right, those dictionaries are wrong. The proof is in the hearing, which you have done properly.

Those sources are "lying" to you. To put a finer point on it, they have committed a certain class of formal deductive error. In particular, just because some people say some instances of certain sounds differently in some words doesn't mean that all people say all instances of certain sounds differently in all words.

Therein lies the flaw.

Aristotle had a thing or two to say about such "reasoning". Then again, you have to be careful with that guy because this instance is just as bad as when Aristotle notoriously decided that women have fewer teeth than men without bothering to check his wife's mouth.

Don't forget to check their mouths.

There is no merger of the interjections because that would confuse their meaning. Each of oh, uh, ah, aw means something different, eh? Those five interjections sport a distinct vocallic phoneme allowing them to be easily distinguished when heard. Each uses a different cardinal vowel such that each is different in both sound and meaning from the next:

1. Interjection uh has ʌ as in cup.
2. Interjection ah has ɑ as in father.
3. Interjection aw has ɔ as in honk.
4. Interjection oh has o as in broke.
5. Interjection eh has e as in very.

Here are their different meanings per the OED, with a single illustrative citation provided for each:

1. uh = An interjection expressing hesitation.

1977 N.Y. Rev. Bks. 4 Aug. 32/4
‘Perhaps we should, uh, wait,’ I said.

2. ah = Expressing entreaty, appeal, or remonstrance; (formerly also) †used to gain attention.

1972 G. Lucas et al. Amer. Graffiti (film script) 40
Ah, Wendy, my old love, come back here and console me.

3. aw = An exclamation usually expressing mild remonstrance, entreaty, commiseration, disgust, or disapproval.

1932 ‘Spindrift’ Yankee Slang 9
Aw! Shucks, forget it—to tender thanks for courtesy or kindness is to provoke this phrase.

4. oh = Expressing (according to intonation) surprise, frustration, discomfort, longing, disappointment, sorrow, relief, etc. Frequently preceding another interjection.

2000 S. King On Writing 186
This did not..keep her from yelling ‘Oh shit!’ if she burned the roast.

5. eh = An interjectional interrogative particle; often inviting assent to the sentiment expressed.

1816 ‘Quiz’ Grand Master vi. 132
What have I brought you here for—eh?

It wouldn't do much good to eject an interjection if nobody could tell what you meant. So yes, these are different. Aristotle should have looked in his wife’s mouth.

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Why ‘General American’ is a myth

Last but not necessarily least, please remember that "General American" is something of a myth, a sort of consensus-reality with little scholarship backing it. Almost every word can be, and is, pronounced nearly uncountably many ways by the billion people who speak English. You have to do real studies, ones that almost always cost real money to put on, to tease out all the myriad differences.

Even this site barely scratches the surface. Take a look at some of their transcriptions for daughter here, starting with North Carolina’s pair of diphthong versions with [ɑˑɔ] and [ɔ̈ɔ̝] as the stressed vowel:

North Carolina [ˈdɑˑɔɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɑˑ     open back unrounded vowel      U+0251  LATIN SMALL LETTER ALPHA
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɔ      open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

North Carolina AAVE [ˈd̬ɔ̈ɔ̝ɾə]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d̬      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
        voiced                         U+032C  COMBINING CARON BELOW
 ɔ̈      open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        centralized                    U+0308  COMBINING DIAERESIS
 ɔ̝      open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        raised                         U+031D  COMBINING UP TACK BELOW
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ə      mid-central vowel              U+0259  LATIN SMALL LETTER SCHWA

Standard Canadian [ˈdɔ̞ˑɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɔ̞ˑ     open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        lowered                        U+031E  COMBINING DOWN TACK BELOW
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

Chicago [ˈdɔːɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɔː     open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        long                           U+02D0  MODIFIER LETTER TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

Chicago AAVE [ˈd̬ɔˑɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d̬      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
        voiced                         U+032C  COMBINING CARON BELOW
 ɔˑ     open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

New York City [ˈdɔˑɾə]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɔˑ     open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ə      mid-central vowel              U+0259  LATIN SMALL LETTER SCHWA

Alabama [ˈdɔ̞ˑɾə]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɔ̞ˑ     open-mid back rounded vowel    U+0254  LATIN SMALL LETTER OPEN O
        lowered                        U+031E  COMBINING DOWN TACK BELOW
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ə      mid-central vowel              U+0259  LATIN SMALL LETTER SCHWA

Boston [ˈdɒ̜ˑɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɒ̜ˑ     open back rounded vowel        U+0252  LATIN SMALL LETTER TURNED ALPHA
        less rounded                   U+031C  COMBINING LEFT HALF RING BELOW
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

Standard US [ˈdɑ̟ˑɾɚ]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɑ̟ˑ     open back unrounded vowel      U+0251  LATIN SMALL LETTER ALPHA
        advanced                       U+031F  COMBINING PLUS SIGN BELOW
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɚ      rhotacized mid-central vowel   U+025A  LATIN SMALL LETTER SCHWA WITH HOOK

Ohio [ˈdɑˑɾɹ̩]

 ˈ      primary stress                 U+02C8  MODIFIER LETTER VERTICAL LINE
 d      voiced alveolar plosive        U+0064  LATIN SMALL LETTER D
 ɑˑ     open back unrounded vowel      U+0251  LATIN SMALL LETTER ALPHA
        half-long                      U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɾ      alveolar tap                   U+027E  LATIN SMALL LETTER R WITH FISHHOOK
 ɹ̩      voiced alveolar approximant    U+0279  LATIN SMALL LETTER TURNED R
        syllabic                       U+0329  COMBINING VERTICAL LINE BELOW

That’s just from North America, too, not the British Isles or elsewhere. Look at their site for the many other possibilities.

A dictionary is only going to include one, maybe two, pronunciations for a given word. And they aren’t going to go into the details the diacritics show, either. What about the millions and millions of people who do not use the dictionary’s choice? Do they not speak English “right”? How would they feel about being neglected?

This high level of variation means that no dictionary is ever going to manage to encapsulate the real variation that exists in the real world. There are all kinds of different influences that change what’s actually said. Do not expect a dictionary to tell you “the truth” about pronunciation, at least not when it comes down to subtle matters as fine as many you have brought to our attention. These take professional work to be seriously answered, not just dictionaries meant for the billions.