If you can do Nashville, you can do Montgomery. Don't do either in the rush hour. You won't find many campgrounds in the Deep South closed in the winter, although you will periodically encounter cold blasts of air from time to time.
The only place in three years of fulltiming, including most of that out West at elevation, that I ever had water freeze was in Gulf Shores, AL. Just proving there are no guarantees!
In my opinion, you may find the Natchez Trace very bleak in the full of winter (although I did not catch when you were going). It is a better late fall or early spring trip, but to each his own.
We normally winter in the Orange Beach area, but this year will only be there in January. It can be blustery at times, but as far as we are concerned, the off season is the most enjoyable time in that area. And if you go hungry for lack of places to eat well, it's your own fault!