I dont see any good reasons to visit Memphis TN , Houston TX or Louisville KY...
NashvilleTN is more for entertainment, like Austin TX, not fur culture.
If you want to explore culture/history, and cuisine, as well as the beauty of the South, I would suggest a not only to focus on the Charleston/Savannah areas, but also, New Orleans, The Natchez Trail area, and parts of Bluegrass Country like Asheville NC, the Smokey Mountains, and also Colonial Williamsburg Virginia, since is played a huge part in the civil war...
Obviously you need to rent a car for all of this, though you can from Wash DC to Charleston/Savannah via train. You could take a river cruise on the Mississippi from New Orleans to Natchez to Memphis...and maybe even St Louis. St Louis played an important role in both the westward expansion with the Louisiana Purchase from the French, it serves as a major conduit for the under ground railroad in the slavery era, and also the industrialization of the west, as wall as the starting point for Lewis & Clark expedition to the headwaters of the Missouri River/Mississippi River and the west to California.
Historic towns/small cities to also consider while on the road would be too many to list...
It depends how many weeks you have for all this.
But Texas is a huge state and very out of the way, and not the Old South. There is no reason to go there IMO.
Adventure Travel to Colombia, Costa Rica, Nicaragua, Honduras, Guatemala, Mexico, Thailand, Burma, Laos, Cambodia, Vietnam, China, South Africa, Morocco, Turkey, EU, USA National Parks, enjoying culture, cuisine, motorcycling, scuba diving, surfing, sailing, rafting, hiking, fishing, camping, nature, wildlife. Get a Guidebook, and get lost!