Might be pretty warm in the Davis Mtns., so outdoor activities during the middle of the day might be not so much fun.
I'd head to higher elevations. But it's a tough week to get reservations. Here are just a few places I found you might still get in. You might have to piece together several different places to fill up a week.
East of Pagosa Springs. Check out West Fork NFS CG. Still had some sites available for that week, but no hookups there. (Right nearby is Bruce Spruce Ranch which hs some FHU. We have driven by but never stayed there.) Pretty nice temps up in this area during the middle of the summer (you'll often need a light jacket in the evening). Head up to Wolf Creek Pass area for great hiking and wildflowers. Go to Pagosa Springs to take in the pool or take a day and tube down the river.
McPhee Reservoir has a CG that rarely fills. Near lots of popular places like Mesa Verde, Hovenweep, Durango, Telluride. There are a few spots there with FHU. I noticed that site 29E is available from July 2nd through the 10th. Enough to keep a family busy for a few weeks.
Often there are sites up at Mesa Verde at Morefield CG. Some FHU sites but those book quickly. Being up on the Mesa means being cooler.
Trying to find something during the week of the 4th of July will be tough at this late date. If you can go w/o hookups, it really opens up the available spaces. Check recreation.gov for what might be available. Look for places away from the main tourist draws such as Durango and you'll have better luck.
Just to say it again, it will be much cooler in the mountains of CO or NM.