Well - while it does sound suspicious, just washing your own sheets is not absolute proof he's sleeping with her there.
If you want proof so he can't gaslight you when you tell him he can't have her there - get one of those Nannycams and set it up in his room. (This might not be legal in your state though.)
But really, you already know he's cheating with her, and if you were in a better financial state you would probably make him move out, right? If you're not in a place to do that, you need to figure out how to get to a place where you can. And meanwhile, just stop and ask yourself about any action: "Will this get me closer to my goal?"
