Sorry, OP, I think it’s best for you and the children if you separate before it drags you down further.
I’ve had a close-up view of the 20-year marriage of a couple where the DH sounds very like your husband — a kind, gentle and intelligent man, but longterm depressive, low-energy, friendless, unable to cope with his own kids for more than very short periods without explicit instructions, meals left etc, easily overwhelmed, deeply self-pitying, sleeps a lot and is otherwise on the sofa, is disastrously disorganised and forgetful, and makes heavy weather of the littlest thing — his wife carried everything to do with the house and children and life admin while working FT, and ended by having a near nervous breakdown herself.
Once they had agreed he would move out, he did nothing for months until his wife actually started looking at nearby rentals for him, and then he just kept postponing the move and saying he couldn’t cope with buying basic furniture. I don’t think he would have left at all in the end if she hadn’t gritted her teeth and organised connecting utilities and buying stuff. Even once he’d moved out, I think he thought she would be coming over to put together flat pack furniture.
Since they separated, her life has been far better, but the agreed 50/50 residence has turned out to be more like 80/20.
But she wasted 20 years trying to prop him up at the cost of her own MH.