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.

You definitely deserve to be at least a co-author. In fact, if I have understood the situation correctly, you deserve to be the lead author with the professor as the co-author. It does not matter that you are paid by university. That does not in any way go against your right to receive credit for the work you have done. Instances of senior researchers using their position of power to take undue advantage of their juniors are not new in academia. For junior researchers, it is very difficult to take any action against their seniors, more so because academia runs on a strongly hierarchical system.

However, you should definitely try to at least talk to the professor. Politely explain that you feel that you deserve co-authorship. You can refer to the ICMJE authorship criteria and specifically point out the areas where you have contributed.

I have recently written an article about such kinds of unethical behavior that is rampant in academia. You can read it here.