So sorry working full time and juggling childcare is a nightmare. Having to deal with insurance and security is also a nightmare as well as an emotional experience. Do not be a martyr to the cause your children have two parents and even if their father has no home of his own it does not prevent him taking them out for the day or a few hours even or taking them to grandparents. It also doesn't prevent him helping out with childcare on occasion. It is important that he doesn't lose touch with his children and become an absent parent unless there are good reasons to keep him distant.
I have been an emergency childcare option for a parent at the local school who worked and if her children were too sick for school I would look after them. I also am the pick up for some children at school and look after them for a few hours until their parents finish work. Networking with other parents can be a godsend if you need one off arrangements in an emergency, formal childcare in these situations can be impossible. I too had no parental support to call on and no witnesses would help so had to work out strategies that worked for me.