Am I being "too sensitive" because I don't enjoy being bullied and called names?

I think its completely inappropriate for anyone to call you those vulgar names! I'm shocked that your manager is refusing to do anything about it! I think you should make it clear to her you will not tolerate such names even if she is just kidding. Put your foot down and let her know you will take it to HR if she doesn't stop. And if she doesn't stop and HR doesn't do anything about it then I don't see why you'd want to continue working their anyway...