I hate to say this but your L doesn't seem like the sharpest tool in the shed either that or he's such a genius that I don't understand his logic lol
Signing over something without getting something in return is NOT how the negotiations work. It's tit for tat.
Also, don't just ask for something from the house either. Use the house as the bargaining chip for EVERYTHING you want. Tell your L:
"I will surrender all interests and debts in the property (Address: blah blah) to be solely H's. In return I will take a lump sump payment of $xx in cash and I will keep my car (make/model/year) free and clear etc"
THEN you deal with the support payments as a separate issue.
Me: 35|WAW: 38|D: 6yo | http://tinyurl.com/2dxx7m6 Feb 2006, left, came back in two weeks Aug 2006, left again Apr 2007, filed for divorce Dec 2007, reunited Mar 2010, moved out, filed again