Good point about the weather. Note that the Joshua Tree National Monument web site has the elevations listed for all campsites, so if you do not want to use AC, you might choose a higher campground.
All of the campgrounds that are developed at Anzo Borrego are in the same area. I stayed at the Borrego Palm Canyon campground. Nice view, and they even let me wash my RV! Or at least when I asked, the camp host told me there was no specific rule against it. With water shortages, it might be different now, however. There was a very nice paved sidewalk path from this campground to the visitor center through the desert where they had signs identifying plants and such.
FYI, while there are no large stores in Borrego Springs, there is a small, but nice, grocery store in town.
This is the area, by the way, that has all the sculptures in the desert. They are spread out all over, but you can park along the main road and walk to them.
http://www.desertusa.com/borrego/bs-art.html