While Mustang Island is indeed outside Corpus Christi, Padre Island abuts Mustang Island out there. In fact, Padre Island National Seashore is right there.
You can't get to South Padre Island from north Padre Island near Corpus via the island. You have to go inland, and it's a 175-mile drive.
Adding to the confusion is that the highway in Corpus that goes out to Padre Island is called South Padre Island Drive, but the "South" means it heads south; it has nothing to do with "South Padre Island."
People who mention "South Padre" and "Corpus Christi" together (it happens all the time) need to clarify which one they're talking about. As ronspradley noted, there's a wide variety of RV parks in Corpus; the most resort-y ones are in Port Aransas, though.
Down in the town of South Padre Island, near the southern tip of Texas, there are 3 choices on the island. There are two county parks: Isla Blanca (enormous and catering to snowbirds) and another whose name escapes me and is just a parking lot with lines drawn on it. And there's a big KOA right by Isla Blanca.
Port Isabel, the town on the mainland right there, has a few RV parks, typically full of snowbirds and other relatively permanent residents.
The weather in both Corpus and in South Padre Island in February will be humid, probably cool, and certainly windy.