We are thinking of 7 to 10 days in the Santa Fe/Taos area
First of all let me say, I think you have made a very good choice for a visit to Northern New Mexico. This is one of our favorite area in the entire country and we go back there almost every year.
I think June might be a very good time, but even late May might work out just fine.
We prefer to say in "Santa Fe Skies" which is a commercial campground. We like to have full hook ups and free WIFI, but there are some forest service campgrounds on the road up to Santa Fe Ski Basin.
You could easily spend a week in Santa Fe itself. We alway enjoy a drive on "The High Road to Taos" with stops in Chimayo, Truchas, and Trampas including visits to the weavers and the Sanctuario de Chimayo. One of our favorite restaurants is called "Rancho de Chimayo." We always try to plan a lunch stop there on the High Road.
We also enjoy the drive out to Pecos. Our restaurant out there is called "Frankie's of the Cassanova." Don't miss the green chile stew at Frankie's. While out there you can visit the Pecos National Monument and take a drive up the canyon toward Cowles and the Pecos Wilderness Area. For that matter there are several National Forest campgrounds on that drive.
We usually stay at "Taos Valley RV Park" for the same reasons listed above, but I believe there are some National Forest campgrounds on the road up to Taos Ski Basin.
There are two many sites worth a visit in both Santa Fe and Taos for me to try to include them here. The point it you have made an excellent choice for your visit to New Mexico.
I also agree with the comment about Chama, and if you go up there from Taos there will be some nice forest drives after the desert around Taos. You could also go on up to Pagosa Springs if you had another week. That is a beautiful drive from Chama up to Pagosa Springs.
If I knew your route I could probably make comments about campground along the way. We've been making that route from East Texas to Santa Fe and Taos for many years.