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

Solution 1:

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.

Solution 2:

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