A few more suggestions (Tonto & the Superstition Mountains are both great):
Look into camping at Dead Horse Ranch State Park, which is about 30 minutes from Sedona and close to Tuzigoot & Montezuma Castle National Monuments. Hike the Red Rocks area near Sedona, scramble up Cathedral Rock and explore nearby trails, see ancient structures on the Island Trail in Walnut Canyon, drive 89A to Coconino National Forest, where I recall some decent dispersed camping sites. Do you know that the Sedona McDonalds is the only one whose arches aren’t golden?
Wupatki and Sunset Crater are just north of Flagstaff, stop by on your way to the Grand Canyon. Get on reservation.gov now and look for a campsite at Mather, which is a good location from which to explore the South Rim. Drive out towards Desert View, hike the Grandview Trail, or maybe down to Ooh Aah Point. Catch the red shuttle to Hermits Rest and walk along the paved trail for beautiful views of the canyon. If you’re adventurous, hike down Bright Angel Trail to Indian Garden and back. If you start early enough and want to work a bit harder, add on several extra (mostly flat) miles to admire the view from Plateau Point.
Consider adding in Petrified Forest National Park If you have time, it’s further east.
Have a blast!