He said he's seeing someone new

[4ever_Regretful]

My loyalty is to you and not your H, so please bear in mind my suggestions are bias to you and your well being.

Even if speaking to a lawyer to obtain some clarification on the implications on what the proceedings entails would be beneficial. Don't make a decision without all the facts.
As for upsetting your H, let him be. What if he does come out of the fog and regrets what he has done? At least you speaking to a lawyer may allow you to figure out a way to protect yourself or at least buy you some time for him to come around.

Good DBing right now is to not have contact with him and not put any pressure on him, but what if I need to do just that in order to end up with a fair deal, or even just to find out some more details about the deal I've been offered? What if I try to get some of the conditions of the settlement changed, and he ends up changing his mind and decides to take the furniture/whitegoods off me instead, leaving me with nothing?

Either way, would you want to go into the agreement blindly?
I think it would be in your best interest to find out the details. You can still decide if you want a letter from your lawyer to be sent to your husbands.

I wish I could just pick up the phone and actually have a conversation about all of this with him, because it's so unnecessary to be making this all so complicated by getting lawyers involved. But I know that he's not thinking rationally right now, so anything I suggest he'll disagree with just on principal, so trying to talk sense to him will only push him further away, (not that he can actually get much further away than he is already).

Not calling him may be the best. I agree even if you speak to him it may not be productive.

Part of me also wants to discuss it with MIL, because maybe she could talk some sense into him and get him to drop this settlement thing. All he wants is the freaking pool table for crying out loud! That's what the letter says, I get to keep all the furniture/whitegoods, but he wants the pool table. Doesn't he realise that if he wants to take the pool table, I'll let him take the freaking pool table! We don't need a legal document to say that he can take the damn pool table!

Don't involve MIL. Having her speak to him would be worst than you calling him directly. He may feel that not only are you disagreeing, you are getting allies to your cause.

If you feel he is simply doing this for the pool table perhaps you can let him know he can have that without the paperwork signed. But there may be something bigger in his mind he is hoping to accomplish.

What am I gonna do? How can I possibly balance sensible DBing with getting lawyers involved? If I go see a lawyer, I'll have to decide then and there how I'm gonna respond to the property settlement request, and I honest to God don't know what I'm supposed to do.

Perhaps you won't have to decide there and then. Before going to the lawyer perhaps you can get a counsel over the phone and determine if you can just go to get the facts of what it all means.

Wishing you strength and courage.