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