I find curiosity very useful, at least when it comes to thinking. When it comes to dating, that is an entirely different story. If we didn't end on good terms, I tend to question, why would this time be any different?
She was going to start sending me a check monthy and asked for my address. When I gave her a PO Box she then changed her mind that she couldn't mail me anything that month because of unexpected bills for her dogs. A few days later she said she could pay as it turns out, but... it had to be in cash so could we meet up?
That freaked me out a bit and I never responded back to her ever again despite several text messages from her in the following days and weeks.
She wants to get back with you.
If she was interested in just paying you back, then she would have gone with the PO Box. So clearly she wanted to meet with you.
Most people would certainly not have bothered to continue with lots of messages, because she's the one doing you the favour by paying you the money back. So clearly she doesn't want to take no for an answer.
Also, she asked for your address, when she could have just asked for bank account details, or a PayPal account, and made the arrangements via email. If she would have rather spoken to you directly, she could also have asked for a phone number, and made the arrangements that way. If she wanted to ensure you got the money in person, she could have simply arranged to meet up for 5 minutes every month, on a pre-arranged basis. She was clearly using the pretext of offering you money, for a way to get your address, because then she knows where you live, and then she can just turn up any time she wants to see you. She was being manipulative to put herself into a situation of control.
Clearly, she wants to meet with you, and she doesn't want to take no for an answer, and she wants a situation in which she can drop round any time she feels like. Friendship is an open thing, 'I'll see you when I see you', sort of thing. This is far more invasive than that. So I'd say that she definitely wants some kind of a relationship. Given that she wants to pay you monthly payments, for what I would suspect would be a long time to pay back 4 grand, she wants an ongoing long-term relationship. Ergo, she wants you back, and for an LTR.
At least, that is what she is currently telling herself. Whether or not that would be something she can handle, is another story entirely.