I agree to see a professional, like a credit counseling type of place, to help you straighten it out. And even more importantly - do you already have a lawyer (I know - no money - but you can usually find one to do a free or cheap consultation!) for the divorce process? You need one ASAP! Don't try to untangle all of this alone.
"He thinks it's not necessary to address this in the separation agreement." Oh, I beg to differ with him. It is most necessary, and all separation/divorce agreements address financial aspects! It doesn't matter what is in his name; as a married couple, you will be entitled to about half of "his" assets, from things like the sale of the house (if you sell it), to a portion of his 401k. And if he makes more than you, he may be entitled to pay of some of your debt, as well as at least a portion of legal costs. Also yes, go ahead and ask that your child's health insurance be covered by him - or at least half of those costs - that's a very common agreement.
Get professional help with this, don't wait. I have a friend with kids who's going through a divorce but her stbx is still living in the house - and the charming man decided not to give her any grocery money or household expenses. She got a court order that stipulates he has to give her something even though he's right there and not even divorced yet. So don't let your ex tell you what's what!