My ex was depressed all thru our marriage, no matter what I did.
He was on antidepressants, but took no responsibility for his moods and behaviour.
It was so hard for me, pretending to the kids that everything was OK, not being valued for all support I have given him and so on.
In my ex's case it was almost like chronic state of feeling sorry for himself.
Heard the term used - learned behaviour, which in his case is/was letting others to feel sorry for himself as if that was enough for him to wallow and be excused from actually adressing his issues.
Apart from the actual medical cases we are responsible for our behaviour and mostly moods we are in. That's why councelling works, mild medication does too.
I said to my ex on few occasions, that he would not treat his workmates as he treated me. So I see no excuse for his behaviour.
Also at the beginning of our relationship I told him that if he turned an alcoholic I would kick him out.
I think if your DH wants to make soem progress tell him to concentrate on improving relationship with his kids, even though yo uaren't going int odetails I am almost sure is not good.
He has to find time to go during weekend and spend time 1-2-1 wit heach of them.
Thatis his duty and doing stuff would give him something to enjoy and focus on.
As for foul moods - don't tolerate it.
Separation under the same roof gives people breathing space. Please consider it. It's not about splitting, is about evaluating what matters.
In case you decide to go on your own - he already shown he's hoing to support you. If being alone and feeling sorry for himself is what he want's - so be it.