You can do one thing this month and change your mind next month. You don't have to make promises or commit to anything you are unsure of.
You could put that house payment under your mattress and then if you get back together you'd have extra money to catch up on the mortgage. It will be quite some time before they actually take the house.
Is it possible for him to pay the house if he makes it a priority? Work more, prioritize spending, get a job, things like that? If that is possible then I personally would let him make it a priority and if it isn't - then it just isn't but you found out something about his priorities too.
ETA: in my state without legal documents in place any debt my husband racked up was automatically half mine - regardless of if we were living together or not. These are types of things you need to find out.