There are dozens of campgrounds in the Rio Grande Valley, which runs from Mission, TX to Brownsville. Last March, we stayed at Bentsen Palm Village in Mission. It was very nice and reasonably priced. It is near Bentsen Palm State Park which is excellent for bicycling and birding. The thing we didn't like about the RGV is it was very windy (20-40 mph) almost all the time, even at night. The other problem is when we inquired last March about reservations for January and February of this year, they were already booked up with a waiting list.
Many snowbird campgrounds are hard to get into for January and February, but if you check campground listings for Mission, McAllen, Pharr, Weslaco, Donna, Mercedes, and Harlingen you are bound to come up with vacancies. Check the iRV2 campground reviews or rvparkreviews.com
for reviews of some of the campgrounds.
You also mention liking the beach. The RGV is near South Padre Island, which also has campgrounds nearby. When we stayed in Mission, the TV stations out of Brownsville also gave weather info for South Padre and we noticed the temperatures there generally ran 5 to 10 degrees lower than the RGV (good in summer, not so good in winter). You will also want to avoid South Padre in March because of Spring Break.
Good luck in your search.