Sister in law probs

My sister in law made up a story saying I called one of our friends cheap in regards to a party they were throwing. I never did, the thought never even crossed my mind. She told the party thrower and all of our family. I think she has convinced herself that I was insinuating the person was cheap. I’ve went over my words many times and with other friends. She also accused me of talking about her to this friend, which I have never done. My sil has been a problem in my marriage for years.

My game plan is to be polite at family parties/events and ignore her otherwise. I say as little as possible in front of her because she always takes everything the wrong way. The situation has caused me to really self doubt myself and how I come across. I also resent my husband for not telling her to f off. He believes me but wants a relationship with his sister. His argument is “who cares, everyone knows how his sister is.”

Anyways, aita for wanting to avoid her at all costs?