Here are a few suggestions along your intended route of travel.
At Williams, AZ you are less than 60 miles south of Grand Canyon National Park
East of Flagstaff, Meteor Crater and Petrified Forest National Park
In Amarillo, the American Quarter Horse Hall of Fame and Museum
In Oklahoma City, the National Cowboy and Western Heritage Museum and Oklahoma City National Memorial
In Memphis, Mud Island has history and model of the Mississippi River and of course Elvis and Graceland but I'm guessing you already knew that. There is also the National Civil Rights Museum
Louisville is near the northern end of the Natchez Trace Parkway and the point where Lewis and Clark linked up in November 1803, prior to the beginning of their expedition in 1804. The Clark cabin is across the river in Indiana and an Expedition Visitors Center is also in Clarksville, Indiana, just across the bridge over the Ohio River.
In Lexington, KY about 70 miles east of Louisville, the Kentucky Horse Park, might be a good day trip from Louisville.
In Natchez, MS there is a pilgrimage each spring and fall which includes some extraordinary house tours. Riverview RV Park across the Mississippi in Vidalia, LA is right on the river and you can sit on the levee and watch riverboats and barges go up and down the river.
In Galveston, the Texas Seaport Museum is worth a stop. Galveston has some great beaches.
South of Galveston on US 59 to Texas Highway 35 is Palacios a quiet town on Palacios Bay is a nice town on the Gulf of Mexico, the RV Park we enjoyed has closed, there is another listed. The city is worth a few days stay, walk the park along the bay and take a rest from travel. There are fishing boats going in and out of the harbor and some good seafood in local restaurants.
Take TX 35 on south to Aransas National Wildlife Refuge. It's a great place during the fall bird migration, see birds, alligators and more.
Corpus Christi has the USS Lexington, a WWII Essex class carrier. There are tours and reenactments and nearby restaurants. There are numerous campgrounds in the Corpus Christi area. Enjoy the beaches in this area and explore Mustang Island for some fun on the beach with a golf cart or 4 wheeler to aid your exploring.
In San Antonio the Riverwalk is one of the prime attractions. Enjoy walking along the river, there are restaurants, boat tours, shopping in the small shops and art stores of La Villa and also in the Mall. The Alamo is right near the Mall as is the Menger Hotel where Teddy Roosevelt set up camp in the bar to recruit troops for his Rough Riders. The Mission Trail is worth exploring. San Antonio is a very large city today and there are RV Parks in the city but if you prefer not to stay there, Boerne is a small town about 30 minutes north of town on I-10 and is a good base for exploring the city.
It takes travel on smaller roads to get from San Antonio to Big Bend National Park but we certainly enjoyed spending a week there several years ago. Big Bend is one of the least visited National Parks in the US and most visitors are there in the spring so a fall visit should be quite relaxing and enjoyable. There are great trails for all levels of fitness as well as 4WD roads to explore if you are equipped for those. Nearby is Big Bend State Park which has campgrounds and is itself worth exploring.
Going on to Roswell, NM you will be near Carlsbad Caverns, one of the most spectacular caves in the US. You can walk into the cave from the surface, a wonderful experience, or take tours that take you into the cave via elevator.
Enjoy your trip and don't forget to come back here and tell us what you did and how the trip went. You might even want to start a blog on your travels.