I'm really sorry to hear that. It's always hard to find out that someone we trusted has betrayed your trust. I think you are probably right that your sister has some sort of problems she hasn't worked through.

Taking time off from her sounds like a good idea. I would have a wait and see approach to cutting her off entirely. Maybe in a while you'll be able to communicate with her again and convince her to seek professional help.

If I understand correctly, the lie she is telling about you is that you go on and on about your own problems and don't listen to hers? If so, other people are probably not going to pay any attention to that. It sounds more whiny on her part than anything else.