Green cards and SSN's are not something that you just turn on and off depending on where you wake up January 1.  You need to refer both of them to specialists in international tax, and immigration law.  

And then there might be community-property issues -- after all, most of our states inherited those rules from the Napoleonic Code.  Nowadays France seems to have choices concerning the "matrimonial regime," which is an interesting way to describe financial aspects of a marriage.