I can relate to a lot of what you are saying here, i really can she says surveying the bombsite that is our house and it can certainly be a bone of contention with us too. I understand the total lack of motivation when it comes to the house.
The thing is, you are taking on so much BLAME here and its not right.
You are overweight - so am i, hugely, but i don't care (despite having my arse flammed to eternity and back for daring to say so on here) I do care about my appearance but am not happy with it. I have very little confidence.
I know what you mean about school hours jobs - i am suffering the same plight just now (i thought it was supposed to get easier when they went to school!)
When your DH goes to bed, is he taking a phone/iPad with him? Are you sure he is off to the gym every weekend? I really would want this question answered.
He has said he wants to work at it, so work at it he must - it is NOT OK to drop this bombshell and then expect to carry on like you did before. Has he retracted his statement? My DP did, shortly afterwards, once he realised the rammifications.
Maybe you could both visit your GP, I woul insist on some sort of therapy - it sounds like he is fighting some long term demons. Men are so bloody stubborn when it comes to counselling and medication. My DP would take meds and hve counselling over his dead body, i know that if i didn't have these (waiting on counselling just now) it really would be over my dead body.
Can you take the children out together?
The only time DP and I get together alone is 2 hours every other week when DD goes to pony club, I cherish those 2 hours and sometimes we just sit in the car and talk, but we are alone with no interruptions.
As for the arguments about the house - i would suggest
a) get a cleaner if you can afford it (i wish!!!)
b) maybe get him involved in a bit of a spruce up togeher, have a spring cleaning blitz so that there is less to do? Maybe decorate a room?
Really, he needs to work with you on this, he really does - it is cruel to tell someone you dont love them and then expect them just to carry on.