I hear ya, Panda. Charity is also often linked to churches and places of worship and benefits tend to be in the form of tax breaks for working parents and state programmes to allow children of indigent parents to obtain medical care rather than cash payments and limited amounts of assistance with food in the form of grocery cards with restrictions on what you can purchase. Even if you could start off on your own there, which you wouldn't be able to due to visa restrictions, there's zero way you would be able to stay at home with your kids and not work unless your husband agreed to fully provide for all of you financially and probably provide health insurance for the kids, too. There's no income support because you have young children, you would have to work, which again, you won't be able to do until after a not insignificant time there as his dependent.
Panda is 100% spot on, that's how it is for citizens, but for foreigners who have no legal right to be there? The best you'll do is a flight home, without the kids.
And the custody laws are such that, well, I have two friends there, Americans, of course, who divorced other Americans and they are stuck in another state they don't want to be in because they cannot move out of that state with their kids. I have another two dual-national friends (US/UK) who are stuck in the US due to marital breakdown and joint custody. They're free to move about, of course, but they can't take the kids who were born in the UK and also UK/US nationals. It's not like here where the mom can just take off with the kids and get benefits - the custody laws there usually favour both parents doing 50/50 custody and the parents providing financially for the kids.
If he tries to tell you all this information is bullshit, well, he's lying. And you can't trust any agreement he makes because he can renege on them without any repercussions. Even my h wouldn't think of going with me to the US permanently (not that it was an issue) until we were married long enough for him to be eligible to enter as a so-called green card holder in his own right and I don't blame him - I could have just dumped him with no recourse and he'd have had to leave without his kids, who were born in the UK but are US/UK nationals. Similarly, I cannot decide to leave the UK and go and live with our kids in the US without his permission. It works both ways! I even carry a notarized letter from him with permission to take the kids to the US on holiday.
I'll bet if you Google, you'll find forums full of people stuck in countries due to custody laws and/or facing removal without their kids.
Don't be one of those people.
And again, I say this as someone who loves my home state of Texas and my family there. I would never, ever have dreamed of putting my husband in a situation that your h is wanting to put you in because I love and respect him and want him to feel safe and secure.