I would insist that she finds a way to financially take care of herself. This will show you if she's actually interested in being in a relationship with you or not. I also notice a long history of dishonesty. That's going to be a hard one to fix. I'm sure at this point that you're having a hard time knowing whether or not to belive anything that she tells you. It may very well be the lack of trust making you miserable and not the fact that she's polyamourous. If things are so unsettled that you never know what's around the next corner how can you find that solid ground to come back to to focus on your own relationship?
I'd wager that a month apart really wasn't long enough to figure out what it is that YOU really want. I think some reflection here is in order. Think about what you want, verbalize it and be willing to act if this partner cannot meet your needs. This means being very specific about what it is that you need is. She might surprise you and be able to give you what you need if it's spelled out specifically, but if she can't then you know that and you can make the desision to move on if you have to.
Everything will be ok in the end. If it's not ok it's not the end.