Unfortunately, I think this issue goes beyond not being good with bills. (
And, no, it's not true that men in general are not good with money, so you shouldn't even use that as an excuse/justification for his behavior.)
Your husband refuses to share ANY information regarding his finances with you. There is NO sharing of resources. Under the eyes of the law, you two are one household and his money is your money and vice versa, but your husband sure isn't operating like that's the case. Like someone said, it's like you two aren't even married. Shoot, most roommates have more openness regarding their finances than you two.
And worse, you have no idea what kind of additional bills your husband may be running up without your knowledge. Even if he doesn't share that information with you, you are still going to be on the hook for it legally.
I hate to sound negative, but it seems like you are glossing over the magnitude of the problem. This is not a "normal" problem that newlyweds deal with. This sounds like the foundations of your marriage are on very shaky ground and its only going to get worse as the years go on. You taking over the bills is not going to solve the root of the problem here.
You talked about staying with him for the sake of the kids, but in some cases the kids are better off if the parents go their separate ways if the marriage is trouble. This is a terrible example to set for your young children and could hurt them in the long run.
I'm usually on the side of trying to work things out, because there are always two sides to every story, but in this case, the facts are so bad that I really don't see any room for negotiation here. Just being honest.