Hello, have you tried going on council register for housing? If you live with your ex you wont get much if any help as you are classed as living with him still. UC is an arse and takes all income into consideration.
I was thrown out by my ex with our daughter i went to my mums then my friend rented to me but i was lucky, do you have any parents you can turn to or friends as if you take the kids then the council have to house you.
If you go on UC you will get 85 percent of your child care back so can then up your hours to save for some where better later down the road?xx