please advise again

what do you mean ? you are saying your cousin's husband was talking to her on the phone.... are they living separately? give more details
if he is currently not in the same country/city as your cousin he needs to get his butt where your cousin lives, so that the whole legal process of the house transfer takes place in the presence of lawyers and proper officials.

also, i am really worried for your cousin, i think she seems to be a simple innocent woman, i am worried, that if she doesn't do everything right she might even end up loosing the house....i really hope she gets in touch with some sharp lawyers or social workers now, to make sure she gets the house NOW before the husband run offs ...