I have the same problem.In order to get a new passport with my new name I have to register the marriage in Romania. I can do this if I go to the Consulate in New York/Washington DC or I can do this in Romania. You cannot do anything online.
The process is the same like in Bosnia. We also need the Apostille.
It's very complicated and expensive. So I decided that I will wait until I get my american citizenship to go home(5 years) and register the marriage and get a new passport in Romania.
You can go home with your green card and the old passport if it's still valid and the plane ticket should match the name is the passport.