Are you in northern Texas or in southern Texas? That would make a difference on what interstate you would take east. Have you already planned your route? Looking at the map taking 20 all the way over to 26 would be the most direct. The only RV park I can recommend is Tannehill Iron works state park in Alabama just west of Birmingham. We have stayed there 2 times and enjoyed it. nice park with full hookups.
RV parky shows a couple of parks in Flat Rock but I can't say anything about those. Never been there. Been to Hendersonville though and it's a great town. The whole area is great to explore. You will be just south of Asheville so think about going to see the Biltmore Estate there. You didn't say any timeframe for your trip or what months. Fall is beautiful all through the area.