In the Dixie NF, Red Canyon campground, east of Panquitch and west of Bryce Canyon NP, is a personal favorite. It will be warm but the tent sites tend to be nestled under some good size pines and are well shaded. Trail-heads to Buckhorn and Golden Wall trails are nearby and these trail are super. However, Navajo Lake and Spruce cgs (east of Cedar City) are nice and will be much cooler. The Uinta NF has some nice cgs like Payson Lake cg, east of Payson, and Diamond cg, east of Spanish Fork.
As to whether to try to do all the NPs this year or not, a lot depends on what you want out of the experience. We spent a week in Zion NP before Fred and I felt we had really discovered what that park had to offer. Bryce took one day. On the other hand, we have been back to the nfs on several occasions and still feel there is much more to see, do, and discover. But I am a little bias