I agree that she does sound like a user. So you shouldn't feel too bad about sticking around a little longer until you have what it takes to move out. Can you get a roommate situation, a place that will take a dog, on your current income? Have you checked the ads to find out how much you would need to make every month? (Just don't let her find the classified ads with all of your red circles around the ones you like!)
Let's say that you can barely afford to manage on your current income. If you use public transportation to get to work, maybe you can use a food bank to feed yourself. Whatever money is left over, buy dog food--bank the rest if there is any left over.
Do you have family/friends/allies in your area? Most people need a support network of some kind. Are you in touch with any LGBTQ organizations or centers in your area? They can sometimes help you out with emergency housing and the like. In most areas, there are also county resources that you can look into.
You seem to need more income, from what you are saying. So perhaps another job is in order. But I hope you have a separate bank account. If you don't have one, get one TOMORROW.
If you were making more money in the past year or so than you are now, you might qualify for partial unemployment benefits. Or, as someone suggested, you can start lining up school if you are into that and don't have a degree.
A driver's license comes in handy for more than just driving, so you might arrange to get one. But that takes time and energy, and it sounds like you don't have a car. You'll have to weigh the importance of that.
If you have allies who live in another part of the country and you can save enough for basic moving expenses, you might consider moving cross country.
The dog complicates matters, but I know how you feel about giving her up. When my ex and I broke up, I was worried about finances and could see myself trying to live in my car with my cat. I had rescued him from the streets, and no WAY was I giving him up.
