I am sorry you find yourself in a state of limbo OP, fwiw I think it's widespread where there's no issue forcing the pace, ie involvement of a 3rd party, control issues.
Does he have family close by? Perhaps he could move back in with his parent(s), and you stay in the house with your DCs to minimise disruption? If you are close to his family you might still meet up occasionally and take your DCs along like an extended family. That's not going to be feasible long-term though if either one of you meets someone else.
Do you have a supportive family? Are you worried about being isolated, coping alone? Are you currently working?
If they were his DCs I'd suggest he could put them in bed every other night, maybe go on the odd day trip together, attend Parents' Evenings etc, discuss all childcare issues. As he isn't their bio father I feel you need some distance for a while so that you can move on. It could take a while to both reach that point but once you did then there is no reason why you could not have an amicable friendship.
Bluntly if either of you meets someone else soon, it won't be so much of an issue.
Ultimately you know your relationship with any partner will be your DCs' future model of how a relationship ought to be.
It's a scary step admitting things aren't working but how much sadder to stick together and 'make do' for fear of what the alternative is. So many people talk of their ex meeting someone new and settling down very quickly, the 're-bound' relationship often apparently thriving for years afterwards to general amazement. Isn't it more likely that the first relationship had really burned out, and it took a total break and a fresh start elsewhere to overcome that limbo phase?