Emily - I don't think there's anything wrong with that - SO LONG as you MAKE HIM PROVE IT for a LONG time before you get back involved with him.
OW is living with his family??? THAT would be a NO for me. I sure wouldn't date him or sleep with him or spend more than the minimum amount of time with him until OW is completely out of his life.
That being said - this sounds awful, I know - is there any way you could use this to your advantage to get a car out of him??? I know that's one of the big obstacles in your life, not having a car. So - tell him if he's really sincere, he'd buy you a car so you have a way to get to a job and to take your babies around.
Also - give the info on where he's working to children's services. You'll know something important by how he reacts. If he gets made at you for it, you'll know he's NOT accepting responsibility for his kids. That's HIS debt and he owes it, period.
The key here is to KEEP MOVING FORWARD with your life. If you want to leave the door open for him, that's fine - just be very very clear on your boundaries, and make him prove it in every way. If you really just think you're done - that's fine too. Be done. (But maybe you can get a car out of him first??? )