As a full-time paid RA, should I get co-authorship on a paper that I have written?
If you have written the paper, you certainly have the right to be at least a co-author. In fact, from what you said, it seems to me highly dubious that your professor should a an author at all.

Being paid as a RA does not waive your right of getting credits for publications you wrote. Your professor is also paid by the university.

EDIT: 

If you would like to talk to your professor about this, I'd suggest that you talk politely. To be honest, I don't know how to do this, but I am sure this is important (and possible).

Thanks to Captain Emacs for their very valuable advice in the comment.