Take a deep breath and focus on what you can change.
Can you work part time / reduced hours. If your wage is lower Benefits may be able to top up housing and support. I would also speak to tax credits and see if you could still manage financially with fewer hours. CAB may be best for this as they deal with it as a whole instead of individual departments.
I can believe your EX has asked you for money for babysitting HIS OWN children as I have had something similar lately. You need to refuse this and either pay a neighbour or ideally an ofsted childminder, ( tax credits will then cover around 70% of childcare).
I would also look at getting another job with better hours, preferably within the company you work, but if not, what the hell ...kids come first. I know I have taken many minimum wage jobs just so I can make the kids dinner etc..
As I said you could reduce hours, (I know this means wage), but still get a top up from Tax credits, DSS, and even if you own your own home you can occasionally get the interest paid on your mortgage.
If none of these are acceptable options and your DP is good with your kids then stop beating yourself up about providing food on the table and a nice home to live in. Once they are older and have kids of their own they will appreciate it.