Avoidant Ex

Just here to vent and I guess get some sort of clarification.

My ex broke up with me a few months ago. He said he wasn’t happy with us anymore etc. However, he was getting close to this girl we will call Jane. Jane and my ex were getting very close and she would come over all the time. She would make remarks about me, my body, how women „exist to serve men” and basically say I was a bad girlfriend. Came to find out later she was doing this to at least 3 other men in relationships and tried to break them up as well. Unfortunately in my case it worked. A lot of things happened and my ex repeatedly stated he doesn’t want a relationship with anyone and wants to focus on himself however, I keep hearing from those around us that he is on the market and bragging about all these women he is seeing (one ultimately being Jane but she ended up cutting him off). He has since got a new car and when handing me his keys I noticed he put a keychain he bought when we first started dating years ago with my name engraved on it. When we interact he is very hot and cold and will try to do things we did when we dated but then will switch and completely withdraw. I’m confused and do not know how to proceed. I try and make a distance between us however it is impossible to fully go no contact due to our circumstances.

Apologies for any spelling and grammar mistakes!