Depending on what months you will be traveling some Forest service CG close around October and won't til March or April. There are several quite a few Forest Service CG around the Sedona area along Oak Creek on 89A. We have stayed at Cave Springs which is a larger Forest Service CG. No hookups, try to get a site towards the back, less traffic noise. One we loved was Beaver Creek. Only 13 sites no hookups but there are sites that back up towards the creek. Check out this site.
There is a nice state CG not far from Sedona in Cottonwood, Dead Horse State Park, they have water and electric and a dump station near the entrance. Here is a link.
As far as the Grand Canyon, we also stayed at Trailer Village with full hookups. If you don't want hookups Mather CG is nice with larger sites and trees. What is nice about camping within the park is there is a bus that takes you to all the vistas, markets, CGs and village. Once you park you don't have to drive if you don't want to. Parking at the vistas can sometimes be a pain. There are bus stops at the entrance of each CG.
Hope this helps. If you have more questions please feel free to PM me. Enjoy the planning.