I can’t offer much help for the Natchez area, but, while your in that region....consider spending some time around St. Francisville, La. Quite a few very nice Antebellum Homes in the area!
For the downtown area, take one of the golf cart tours, that meet at the visitor center downtown. Be sure and see the movie there. Make arrangements at the visitor center, and drive to the others. Be sure to see and tour Melrose. tours are regular scheduled by NPS rangers.
We visited Natchez a few years ago and faced the same dilemma - how best to view the beauty of the town. DW and I decided to split up - she took one of the bus tours that stopped at various homes, while I chose to do my own walking tour.