For San Antonio, March and November are two of the best months with respect to weather, with average lows around 50, highs in the low 70s. These are also a couple of the driest months, about two inches of rain.
But those are averages, extremes can go beyond. I was in San Antonio for training mid-October through December, and in November we were still having days when high temperatures restricted outdoor activity, and mornings could get frosty enough that we would be in dress blues wearing raincoats for warmth. When it does rain, it can rain hard, 1/2 inch to an inch in a few hours.
Since the early 1980s we've made a lot of March or April (Spring Break) visits to San Antonio and have usually been lucky with the weather behaving within the averages.
As for RV parks, I like Travelers World, just south of downtown on the Mission Trail. While the sites are kind of close together, the location offers easy access by public transportation.
San Antonio KOA also has access by public transportation, is a little further from the center, but closer to the 410 loop, I-10 and I-35 if you want to use expressways to get around by private vehicle.
There are at least a half dozen RV parks inside I-410, at least as many more in the suburbs inside the 1604 loop road (only part of it expressway). Admiralty RV Resort is popular with visitors to the attractions around Seaworld, or folks making day excursions into the Hill County towns. Blazing Star RV Resort is in the same location.
I've not stayed at either, since most of my San Antonio visits were staying with family living there, or staying in hotels near the Riverwalk where my business meetings were. When going in the RV I choose Travelers World.
If you really mean campgrounds, not RV parks, the nearest will be out on Canyon Lake where the Corps of Engineers has campgrounds in the public access areas, or at Guadalupe River State Park.