Finally, what helped us immensely is I stopped greeting him with a "honey-do" list when he got home. I had heard someone on the radio say, "You have to create an environment where he *wants* to come home," and I thought, OMG, for DH, coming home is like the second shift. At work, people looked up to him and did what he said. Why would he want to come home?
this would not fly in my home, but maybe you SAH and can handle it all. I dunno. At work, people looked up to me too and did what I said. Does not mean that I expected my husband to do my bidding, or the other way around--we were a team and needed to act like one. We got home and it was like a military operation: which parent, which kid, which tasks.
OP, this is a very hard time. We had a rough rough time with #2--our toddler (now older and dx wit ADHD ) was a nightmare. Baby was fine, but still it was just neverending at time. DH could be grouchy. But there is a limit. Yes, he may be depressed and incredibly stressed and should be evaluated.
But he also has to realize that this is part of parenting and having a shitty ass attitude about it only makes it worse. And the kids pick up on it and it affects their behavior.
we did occasionally do date nights, get a break, and would alternate taking both kids for a couple hours on the weekend so the other could work out, relax, whatever. We also found other famiilies with young kids so we could socialize at the park,or order pizza and have folks over. that helped with some of the tension of feeling burdened by the family demands.