Did you do this trip, any suggestions for campsites?
Not the OP but we have stayed in the area several times.
There are two developed campgrounds in GUMO - Pine Spring and Dog Canyon. The first is pretty much a parking lot near the Park's Visitor Center and by far the more heavily used. At first glance, Dog Canyon doesn't look like too much either - the half dozen or so trailer/vehicle campsites are loosely defined around a horse corral but there are definite positives. You'll most likely have the place to yourself with maybe a couple of tent campers some distance away. Its in a box canyon with the entrance running just in front of the on-site ranger's home (think security). Can't guarantee this but I've always been able to get good recommendations for nearby local trails in adjoining Lincoln National Forest from anyone on staff there. No showers but flush johns which have been spotless every time we've stayed. A couple of great hiking trails nearby. Its remote, forty five minutes to an hour from Carlsbad but a pretty drive - watch out for wildlife and cattle in the road. McKittrick Cabin is a good half day hike - continue on to the Grotto. Guadalupe Peak is the highest point in Texas if you're in reasonably good shape. Sitting Bull Falls isn't too far off the road leading into Dog Canyon and worth the time.
If you haven't visited Carlsbad Caverns, you should do so and I'd recommend taking the extra cost Lower Cave Tour. Lots of dispersed camping opportunities in Lincoln NF. Brantley Lake State Park is a good reasonably priced option if you have need for a hot shower. Have fun, its a beautiful area.
Edit: If you look at a map, Pine Springs and Dog Canyon appear to be fairly close to each other but there are no roads connecting the two and they're a couple of hours distant.