The southern Utah national parks are awesome and if you don't mind the heat (do siteseeing early mornings) here is a suggestion:
Take 191 south out of Jackson and follow it into northern Utah at Flaming Gorge Nat'l Rec Area to still stay in cool temps. Firefighter's (or is it Fireman's) campground is nice for big rigs - no hookups so come prepared. It overlooks Flaming Gorge.
Continue south on 191 to Vernal (a big downhill) and Dinosaur Nat'l Monument. Continue further south on 191 and now you have many options - head to Green River on I-70 then south on 191 again to Arches National Park and Canyonlands...great Jeeping.
..or at 191 at Wellington, pick up 10 to 72 to 24 at Torrey for Capitol Reef Nat'l Park - many RV parks in Torrey. Then Hwy 12 west to Bryce Nat'l Park - Ruby's RV is at the entrance. Take 89 south to Mt. Carmel and 9 west to Zion Nat'l Park. . . a steep downhill but many RVers do it. Try to stay in the park at Watchman campground (check for reservations) or in nearby Springdale at a RV park.
The list could go on and on because this is an awesome area to tour. You will also be close to Grand Canyon North Rim from Kanab, UT to Alt89 to 64 south. Or Take 89 into Page, AZ and Glen Canyon Nat'l Rec Area and continuing south on 89 (could take 64 west off 89 to the Grand Canyon South Rim) or directly to Flagstaff, AZ will take you to I-40 for your return home.
10 days is not enough time!