First let me say that you are in my prayers and my heart was just breaking reading your story.
I agree with what someone else wrote about calling around for more help. When a similar situation happened to me I called Salvation Army, and FIA which it seems you have already done. Since your hubby won't offer any help to you....call your caseworker and let them know that, they are there to help. They WILL pay your rent so you are not evicted!
I know it is so overwhelming right now, but it will get better day-by-day.

Dealing with an alcoholic husband is NOT easy, I deal with it everyday as well.
Hoping for a better tomorrow for you and your kids.....