Where are you leaving from, South Dakota? I don't see how I-10 fits into this picture?
In southern Texas,
1) Padre Island- major snowbird area
2) San Antonio: River walk, Alamo and the Mission tours
South West TX:
1)Big Bend that would be the perfect time to visit that park and the adjacent state park Big Bend Ranch
2) To the north a bit is Sonoma Caverns-its commercial right off I-10
3) McDonald Observatory if you like astronomy.
4) Ft Stockton-the old fort there.
Various Texas State parks, especially Palo Duro up south of Amarillo. I find their parks are very well kept but most are not spectacular. And for the size of the state they are a bit sparse. Notice that Texas has very little Federal land and I've heard zero Indian Reservations. That last becuase of their history with the Comanches.
Both NM and AZ are packed with State and Federal Parks.
If your rig is small (under 30 ft), we liked Chiricahua National Mon, in SE AZ to be a good spot for a week or so but the CG only takes small RV and we could not find any NF or BLM cg in the area that were not likewise limited. The nearest commercial CG was out on I-10 a long drive off. I suspect that's why you never hear this park mentioned.
Karchner Caverns are supposed to be spectacular but you better have reservations before getting there from what I hear. Organ Pipe NM is said to be hazardous because it is right on the border. Maybe the new fences will return that to usable status. Bisbee is an interesting town.
That's enough from me other will chime in.