First of all, why are you putting what's best for your DH ahead of what's best for you?
You are two adults, with equal importance in your relationship. I suspect that he's good at looking after himself. Does he ever put you first in signficant decisions? Would he put you first to the detriment of what's best for him?
So first of all, put that out of your head.
Next, your children. You assume that the best thing for them is you staying together. And yet they have an unhappy mum whose depression is likely caused by her marriage, and a dad who (from your op) appears to have been controlling, selfish, lazy (albeit temporarily behaving better possibly because he saw his easy life threatened).
Children can happily 'surive' their parents breakup. And in many cases they are better off, in a happier and calmer home.
If your DH reverts to his old behaviour (and in fact you only say he's got better, not that things are great) then your DC will see a poor relationship model, which will likely impact in their future relationships.
I'm not saying these things to make you feel bad. But to show you that 'staying for the children' is often a flawed logic - and often leaving would be best for the children.
SO parking your DH and DC for a minute - what would be best for you? You think you might not be depressed if you were to separate? You're dreaming of another home. Even though he's making an effort, you still think you want to be away from him. And that makes sense after what he's put you through.
You don't have to have his permission to separate. He doesn't have to agree to it or be happy with it. He has put himself first for a long long time. You need to start putting you first. And by default, I think your children would be just fine, and likely would benefit themselves by having their mum back to being herself.
Look after you, and your DC.