I know it's hard watching him get so down on himself, but he had to hear these things. I think if you can keep reassuring him that you love him and don't want to leave him, but these things need to be addressed, it might help a little. But we both know what depression is like and I think we can agree, there's no real good way to bring this up where he won't feel terrible, especially if he's already feeling terrible. Maybe something along the lines of "I know you feel terrible, and I love you, but we have to work on this together as a team, and beating up on yourself is hurting both of us."
What about something concrete, like a to-do list? Not a long one. What I used to do for myself was create a week long to-do list where I'd give myself a whole week to get everything done. It helped because I wouldn't beat myself up every night for not getting everything done. Maybe something like fill out 5 applications, dishes 3 nights/times, vacuum one room. It's not a lot, but it's a start.
But most importantly, keep taking care of yourself and trying to let things go so your stress doesn't escalate too much. Wish I could be more help