I'm so sorry you're in this desperate spot.
First, it isn't your fault. Even if you've done something wrong, you don't deserve to be treated like crap, especially by the person who is the father of your children. Even if he doesn't want to live with you anymore, he's legally obligated to help support those kids.
Next, this is just me brainstorming out loud. Take whatever is helpful, reject whatever is not, no worries.
Once you move out, his income doesn't count when you're assessing your access to social services. And, even with his income, you may qualify for some. Do a search for your state and see what they offer and if there's an office nearby you can visit. In some states, they have one center that has job services and and also can help you apply for additional services like SNAP, WIC, child care assistance, etc. They may have info about subsidized housing in your area. They may be able to give you a referral for someone to help you work out a custody arrangement with your ex, which would also include an agreement about him paying for child support. In any case, go there and talk to someone who works there. It should be a good start to finding yourself and job and a new place to live.
I'm rather disappointed that there wasn't more help for you at the homeless shelter in regards to helping you build a new life for yourself. Another option could be to look for local churches that have a soup kitchen and see if they offer additional services for poor or (almost) homeless people like you. Public libraries sometimes have special programs, too. At the very least, a librarian is likely trained to help you find resources and social services in your area.
Here's a program that could be interesting to you: http://www.coabode.org/programs/program/1
One job to look into: apartment management. I had a few friends who did this and loved the convenience of it. It generally doesn't require a degree (although you need basic office work skills). It provides housing. With some, you can bring your children to work. Or, you may be able to find someone who lives in the complex who would like a childcare job. In some complexes, they may have a separate job for people to clean apartments in between renters and that might be a good job for you, too.
I'm afraid that this forum doesn't get very much activity anymore. You may be able to get more help by finding a local group of some kind on a social media platform?
In any case, good luck!