I'm not a lawyer. I don't even play one on the internet. But it seems to me that your husband wants something and you want something. He wants off the house note. You want the house. The question is one of timing.
The thing about negotiation is that you can negotiate with any rational person on the planet. Really. It's easy.
It's not possible with an irrational person. Believe me
But what you can do is make the offer to remove him from the deed contingent on his payments, his settlement, etc. Somebody has to go first and in this case you are suggesting it is him. When you sign the agreement ask your lawyer about putting in wording to the effect that you'll take him of the note within 12 months of the divorce settlement. His laywer may balk at that and require that you show the ability to get a refinance. What you are after is to buy time while you can get some work history behind you.
Fight for your house. Whatever it takes, it will be worth it down the road. It's not the end of the world if you can't but it's important to you and the kids.
Your Husband won't be any help. You'll have to coordinate around him regardless of his mood swings.
The trick is to get the time and to do things regardless of your anger or how he reacts.
Be careful of emotion when it comes to negotiation. I've been where you are and kept the house. (I wasn't a stay at home parent though; she was before she left.) But staying away from the emotion helped in the long run. I ended up having the house on the market for a year with no reasonable offers. I was not willing to drop the price by much, but it bought me time. A year while we were both on the hook
AJ
"Our lives begin to end the day we become silent about things that matter" MLK Put the glass down... "Yesterday I was clever so I wanted to change the world Today I am wise, so I am changing myself."