She wasn’t being over-sensitive; her boss was obviously sexist, and she did what she felt she needed to in order to have income, including being bawdy with a filthy mouth. No double standard there — that was the culture she was being hired into, and she knew that in advance and accepted it. That doesn’t mean she liked it.
She didn’t expect or ask for Marty to white knight for her; she pointed out to him how he treated her differently because she’s female, he recognized that in himself and also in their environment, and worked to address it since he has privilege that she doesn’t. She needed that because of her boss’s sexism, which she didn’t have any way to fix.
And none of your rebuttal addresses the central point, which was that her work (and Marty’s) was treated significantly differently solely based on the name and gender associated with them.