It is the toughest job in the world - being a single parent. You work 24/7/365... and it doesn't get any easier.
But giving up is not an option. Your children are depending on you and only you and you have to do this for them. To show them that not only can you survive, but thrive!
Start by going back to social services and requesting child support. Regardless of whether or not dad is a loser, he still has a financial obligation to his children and the state has an obligation to assist you in obtaining that support for your children. Also explain to your caseworker that you need transporation assistance, day care assistance and housing closer to the services available to you.
also check out 211.org there maybe additional services in your area to help.
You may be eligible for rental assistance as well - check out hud and see if they can help - their rent works on a sliding scale - which would make it more affordable for you and your children.