well obviously you can't "trust her or this relationship" in the sense you could if she was sane and caring etc. - if this is the way it's going! best thing to do right now is try to minimize the interpersonal conflict and just prepare for your next steps away from her and away from your relationship to her.

save your texts and stuff from her (in case she ever decides to accuse you of something if you break up and won't get back together with her). Just try to avoid the conflict. Right now you're both feeding off a negative social situation and it's just making things worse, so don't get baited into arguments. If you can't leave yet, and she doesn't want to break up, then fine, don't "break up", you can still live in the same place until you are able financially to leave. Just try to minimize fighting completely. Live with her as if she were your sister - if your sister was insane.

none of this is good, she's just trying to push her negative feelings and problems onto you, like she's cheating/on the internet dating sites talking to other guys, saying she doesn't want to be with you, but then when you break up "you're giving up and quitting" - well guess what, you SHOULD BE quitting her, she's not in a mental or emotional state where you can have a healthy relationship. That means that for you and for her, it's best to separate rather than remain in a toxic relationship that brings all this bullshit into your lives. She is obviously not able to have a healthy relationship right now and you're just being sucked into her crazy black hole/gravity well. It's not your fault, you can't change her mentally or emotionally, but you shouldn't subject yourself to constant negativity.