I think that the answer to this question depends ALOT on the view points of the extended family. Is the family of the "biological" mother going to challenge any arrangements for the children in the case that person passes way?? If so, then I wouldn't move to Alabama. The SO would have very little standing here. However, I believe that SO adoption is honored and done in TN so that might be a better option.
From personal experience, finding an attorney that has the experience you are looking for in Alabama is also a tough thing to do. In addition to the "legal" challenges, there are still every day social challenges that exist in the south. I would suggest Huntsville, AL or Nashville to limit "culture shock" between NJ and the south.
Good luck