To sum this up as quickly as possible:
BACKSTORY- DH and BM purchased a home together after being engaged for years and having a child while living in an apartment. Long story short, they parted ways soon after and were never married. DH never pictured not being in the home with his son, however BM refused to try and work things out (family problems, extended family problems, etc). Out of guilt for leaving his son but knowing things just weren't going to work out between him and BM, DH ended the relationship and moved into an apartment. He made it known that they would eventually have to sell the house but he was wanted to work together, he wasn't out to make his son or BM homeless but she needed to eventually find alternate housing. She agreed, but year after year always said she needed more time. I should also add DH was the one to initiate and pay a lawyer to go to court after they split up so he could have parental rights regarding his son. BM was wielding SS like a sword and punishing both for DH leaving, denying visitation upon DH arrival, changing pick up times last minute, and had even left the state on a vacation once before without even hinting that she was leaving.

CURRENT DAY- It's been 7 years now, DH is at the end of his rope. Again, he goes to BM but this time says they are selling the home by the end of this year and she has had more then enough time to figure out an alternative plan. He agreed to give her all equity she has paid into the home, and any upgrades made if they add value to the sale he would also give to her from his share of the sale. The potential profit is enough to allow both parties to essentially pay off debt and put a down payment on a home in their own name. BM response when told this was that DH should just sign over the deed, he owes it to her. When he told her he wasn't just going to 'gift' the house to her that's when things got ugly.
BM was quite convinced DH couldn't sell the house if she just refused to sign anything. DH explained since they were both property owners and not married it wouldn't matter if she refused to sign. He didn't want to do that to her, he would rather work together to sell the home, but if she didn't want to cooperate it would just delay the inevitable. BM then told him that if he didn't sign over the house that she would find alternative care for SS and do whatever it took to make his child support go through the roof, even going as far as threatening to leave the country and start over so DH could never see his son again. BM has extended family out of the U.S. it wouldn't be hard for her to just pack up and leave without saying anything. DH court papers clearly state that "Neither parent can leave the state without OP permission, an itinerary, length of stay, phone numbers, etc". It doesn't say what to do if either party violates the agreement. Does anybody have experience with this??