We are Arizona natives and have stayed at Fool Hollow several times, most recently Memorial weekend. As the summer progresses, Show Low becomes very warm (>95) during the day, so you'll find yourselves inside a lot from about 10 am to 5 pm. We move higher as the summer gets warmer. There are several places near Woods Canyon Lake, such as Canyon Point. No full hook ups, but a pleasant place. They do have some sites with electricity and they have a dump station and showers.
There are several campgrounds at Big Lake (about 9500 ft.) with beautiful weather in the hot summer. Apache Trout campground has a few sites with full hookups.
Also, if you are into just camping in the forest (dry camping), there are several areas to choose from. We like around Green's Peak (about 9500 ft.) and if you can see the peak, you will have cell coverage.
There are also other (non hookup) sites in the forest. Here are some links. We like the Apache-Sitgraves forest for dry camping. There are other campgrounds around Flagstaff that we enjoy.
http://www.fs.usda.gov/activity/asnf/recreation/camping-cabins
http://www.fs.fed.us/recreation/map/state_list.shtml#Arizona
There are also other places around the state, but most cool temperature locations are "dry camping" situations. If interested in more ideas, please send me a private e-mail.