Don't know much about South Padre. Even though I have lived in TX most of my life, I've never been there.
The north end of the island is where the Padre Island National Seashore is. If you are looking to boondock on the beach, I would think you could get a spot there pretty much any time you want. See the following link for more detailed info -
Padre Island National Seashore (U.S. National Park Service)
The beach north of the visitors center is usually a pretty high traffic area and, to me at least, much more crowded.
The beach south of the visitor center goes on for miles (literally). You should have no trouble at all getting as far away from other people as you want assuming the driving conditions on the beach are good. Check at the visitors center for tide information and beach driving conditions. If you get stuck, it can be very expensive to get out. Every time I have been down there though, it has been possible to find a spot within a half mile of the end of the paved beach access road.
If you search for padre island national seashore on google maps, you will see a very good image of the park and visitors center.
Bird island campground which is on the bay side is a little more "developed" in that they have actual defined campsites rather than just wide open beach camping.
A little further north, Port Aransas offers beach camping in several sections of the beach. There, you are a little closer to "things" if you need food, supplies, restaurants or other activities.
There is also a county park at the north end of the island that is just accross the dunes from the beach. I haven't stayed there, but when I have driven through, it didn't look too appealing. It does have a prime location though and the advantage of water and electric if you don't want to boondock.
Nueces County Parks & Recreation Department
Mustang Island State park is about half way between Port A and Corpus Christie. It offers camping with water and electric but it is pretty much - "camping in a parking lot". You can easily walk to the beach though just accross the dunes.
TPWD: Mustang Island State Park
Commercial campgrounds are plentiful in Port Aransas, Aransas Pass, Rockport and Corpus Christie. Beware that winter and early spring are the busy time of the year for those folks with a giant flock of "snow birds" or winter Texans taking roost in that part of the world. Most of those folks aren't really interested in boondocking though so it doesn't really restrict your ability to camp on the beach if that is what you want to do.
Camping on the beach is certainly an experience that you can't get in too many places. I certainly recommend it!