Is there another word for "only giving one side of the story"?

A few points are worth making:

First, find and read Jacques Ellul's magisterial treatment of the whole panoply of historic issues wrt propaganda in his book of that title, Propaganda: The Formation of Men's Attitudes (https://www.amazon.com/Propaganda-Formation-Attitudes-Jacques-Ellul/dp/0394718747/ref=sr_1_1?ie=UTF8&qid=1522928817&sr=8-1&keywords=ellul+propaganda, please forgive the obvious gender bias in his book's title). Written in 1973, Ellul remains the "dean of the deep thinkers" on this topic which has such resonance today wrt the perceived tsunami of fake news.

Second, Alessandro Bessi's papers are among the best contemporary treatments of viral misinformation (https://scholar.google.com/citations?user=47xtEeEAAAAJ&hl=de).

Next, among the many synonyms for one-sided reporting are the ones already noted in the comments as well as terms like cherry-picking the evidence and selection bias.

Selection bias was first studied by Nobel Laureate James Heckman back in the late 70s. His paper introduced a method for quantifying the impact of sampling bias (a subset of selection bias, https://en.wikipedia.org/wiki/Selection_bias, see the last footnote in this wiki article) on research and the need for compensating for this bias in order to obtain more accurate parameter estimates in a statistical model.

Finally, selection bias introduces the whole array of cognitive biases and heuristics that are the research focus of fields such as prospect theory and behavioral economics (e.g., https://en.wikipedia.org/wiki/List_of_cognitive_biases). Daniel Kahneman (along with the late Amos Tversky) won his Nobel for contributions to the field of decision-making and, in the process, challenged neo-classical economic assumptions of symmetric, fully informed, rational choice and replacing that framework with asymmetric (loss averse), imperfectly informed, irrational choice architecture.

Among the laundry list of heuristics and biases at least two are highly relevant to your discussion: confirmation bias:

The tendency to search for, interpret, focus on and remember information in a way that confirms one's preconceptions.

and the bandwagon effect:

The tendency to do (or believe) things because many other people do (or believe) the same. Related to groupthink and herd behavior.

From Oxford Dictionary (online):

Mislead verb (with object) : Cause (someone) to have a wrong idea or impression.

misled [past tense]

• Don't allow the government to mislead the public about the road's environmental impact.

to slant. MW Dictionary v. slant·ed, slant·ing, slants and TFD

to interpret or present in line with a special interest To present so as to conform to a particular bias or appeal to a certain audience.

As in:

The Senator's story was slanted and did not sway me.

The Senator's slanted diatribe could have persuaded many, but not me.

Perhaps the word 'pitch' properly conveys the intend to influence?

A form of words used when trying to persuade someone to buy or accept something.
‘he put over a very strong sales pitch’

Reference:
https://en.oxforddictionaries.com/definition/pitch

Not a verb but a noun - narrative.

I did not let the senator's narrative sway me.

ODO:

narrative

NOUN

1.3 A representation of a particular situation or process in such a way as to reflect or conform to an overarching set of aims or values.

‘the coalition's carefully constructed narrative about its sensitivity to recession victims’

I did not let the senator's personal view…

Less pejorative but necessarily biased point of view as opposed to a consensus or general poll.

Subterfuge. It's an argument technique that employs deception in order to promote one's agenda. One of the most commonly employed is presenting only one side of the story in order to persuade the listener to embrace a conclusion they would likely have rejected had the presentation been more balanced.

I did not let the senator's take (on things) sway me.

Take doesn't necessarily imply any deception and it suggests that there are other viewpoints or angles to be considered other than those of the senator - that this is perhaps one of several versions.

Oxford Living Dictionaries gives the following definition:

take (noun)

A particular version of or approach to something.

his own whimsical take on life

His new takes on the classic tales made us laugh out loud at more than one bedtime.

He is obviously having fun with his take on a midwestern everyman, but make no mistake, it is a take and little more.

A modern version of the same book will have a different take on what we should know.

The main problem with his take on the story is his overwhelming niceness as a director.

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Spin is similar, but implies more of a slant, probably deliberately partial or selective.

Again, the definition is from Oxford Living Dictionaries:

spin (noun)

[in singular] The presentation of information in a particular way; a slant, especially a favourable one.

he tried to put a positive spin on the president's campaign

I think this might sound too consciously misleading for you.

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Both of these answers are nouns, of course; you ask for a verb but your second example sentence can, I think, only be completed with a noun.

Since this word comes up a few times in your question, and I believe it fits best, I think it's worth talking about bias.

I did not let the senator bias me, even though this was the first information I’ve received, and from a source I more or less view as trustworthy.

I did not let the senator's bias sway me me.

Definition:

bias noun
bi·as | ˈbī-əs
1a : an inclination of temperament or outlook
especially : a personal and sometimes unreasoned judgment : PREJUDICE
b : an instance of such prejudice
c : BENT, TENDENCY
d(1) : deviation of the expected value of a statistical estimate from the quantity it estimates

(2) : systematic error introduced into sampling or testing by selecting or encouraging one outcome or answer over others

bias verb
biased or biassed; biasing or biassing
transitive verb
1 : to give a settled and often prejudiced outlook to

his background biases him against foreigners

^{https://www.merriam-webster.com/dictionary/bias}

tendentious

Expressing or intending to promote a particular cause or point of view, especially a controversial one.