I realise you are in a terrible mess and overwhelmed right now, but you have to act, and act FAST.
If this situation is left to drag on, your mental health will get worse, the financial mess will get worse, and your employment prospects will get worse.
First - you are now single. Get your head around it. You live in the same house as your ex, but you are NOT "living together".
So no joint cooking. No joint laundry. A rota for cleaning like flatmates.
Phone the bank and get your name taken off the joint account. Single people don't have joint bank accounts. Any bills coming out of it will temporarily be his problem until finances are settled.
I assume your pay goes into your own current account? If not get this set up asap.
Second - you need proper expert advice about filling in the UC form, as a single person, with no joint finances or joint bank accounts.
Third, put in a claim for maintenance from him, on the grounds that currently you do 80/90% of all childcare, you have them every overnight, they live with you, not him, even though he is living in the same building at present. Again, get proper expert advice with the forms if needed.
Fourth, see a solicitor and get the ball rolling on divorce. I don't know if a solicitor will help with a maintenance claim, so maybe steps 3 and 4 go together.
Fifth, discuss with him whether he will be wanting 50/50 care of the children, in which case which 5 drop offs and 5 pick ups he wants to do every fortnight, and what are his plans for having them all day for 50% of all school holidays? If he can't do 50/50 because of his work, he will have to pay a higher amount of maintenance.
Sixth - discuss with him who will move out, or will you both move out and sell the house. This step doesn't come earlier in the list because you need the finances sorted first.
Finally, when the dust settles and you are each living in separate homes, you can make career plans.