Dual citizenship (Paris)

Bonjour.
I hope someone can help me.
I have 2 passports. For one I need visa to enter France, other is the EU passport.
I married a French citizen and my non EU passport has my new last name. Same as the one in our livret de famille.
I am planning to enter with the EU passport (checked that, all legal since it will take a while to change my last name there).
Now we come to hard part.
Once that I need to apply for the CARTE VITALE I will need to provide proof that I am there 3 months.
I am really confused what passport to use for bills, or anything else except entering the country. If I use non EU I will need to provide proof of Visa, and that I don't need since I have EU citizenship.
If I use EU for all, how do I prove I am his wife?

Thanks