As far as I have ever heard, you either have to have sole legal custody and/or that as the other parent, he has to sign off on them obtaining passports. Sorry. :( I'm betting you're going to have to go through the custody part of all of this first and get it finalized before you can get the passports.
That is really nice that the Sheriff's Dept. is willing to assist. Research the custody laws in your State on a reputable on line site. I think you have a good chance of getting sole custody, especially if their dad isn't getting any help. If you have family who can help you pay for a Lawyer....If you are broke, see if there are any Lawyers in your area doing "pro bono" (free) legal help for women. See if there is a women's services center in your area with help getting free legal help for court involved mothers. Best wishes.