I was there a few years ago in January. It was very quiet and a bit windy every day. The beach is nice, sand is hard near the water and gets softer as you get to the land. I drove about 5 miles on the beach with a truck camper and a trailer and camped there free for a week. I saw a few class C but no class A when I was there. The drive is pretty hard sand for the first few miles, I would think you can go with a classs A, especially for the first few miles, but you can ask at the campground I am pretty sure they can inform you on that. The further you go, the softer it gets. At one point there is a post sign for the need of a 4x4, 5 miles I think, but you can drive quite a few miles without 4x4,it is a nice drive. We drove for about 40 miles and turned around as we where pretty alone and if I got stuck it would have been lonely for a few hours maybe!
We where alone on our portion of the beach, a few trucks and cars going by once in a while.
You can also go to Port Isabel, south Padre Island, there is a campground on the beach too or you can camp on the beach, again, class A not sure. Further down south, follow Boca Chica road to Brazo Island, free camping on the beach too, and you can drive to the end of the Rio Grande, 4x4 needed, then on the other side of the river is Mexico.