I suggest following the gulf coast all the way to South Padre Island Texas, along the way explore Galveston Island, Rockport, Mustang Island, Corpus Christie. At the LA TX border get on the Bolivar Peninsula, take the free ferry to Galveston Island & drive on the Barrier islands as much as possible to North Padre Island. The ferries are free and fine for big rigs, the roads are good. You will miss the horrible traffic in Houston. From South Padre Island head North to San Antonio & then the Hill Country then head west to Big Bend National Park.
