Consider buying your S.O. a puppy. Explain that you are half way through the process and that it will get better. You can start looking for jobs in the area that they wish to move back to. If there is a legitimate reason to leave the program you are at, I would endorse that. If it is just that they don't like living in that city or they feel they have no friends there, that was all known when the rank list was made.
Usually, when there is a legitimate reason to move to another program, only one location is mentioned as a destination. Since you mention N.C. OR Tennessee, it makes me think that it is the city you live in that is pushing them away.
This could be the warning sign you were looking for. Your life as a doctor will involve long hours and difficulties. Having a partner who is supportive will be important. I would make sure that the reason for the move is legitimate (I'm not saying it isn't, it just sounds suspicious). Having a partner who is not on board with temporary sacrifices to achieve long term goals will be frustrating for you. Just something to consider.
Have you considered living a little bit away from the hospital and commuting in? Is it the neighborhood? Is it an unsafe environment for the fiancee? What is it that causes their unhappiness? Is their unhappiness more important than the unhappiness it would cause you to leave your training program (assuming you are happy there, since you pinned the blame on the fiancee)?