A suggestion - consider putting your RV/towed description in your signature so we can see what you have for recommending RV sites. I see from a previous post that you do have a 35' motorhome so going by that:
Zion - stay in the park at Watchman campground; electric and water fill/dump station
Bryce - in the park - dry camping or just outside the park at Ruby's complex.
Capitol Reef - in the park - dry camping or a handful of full hookup parks just outside
Moab - in the park - dry camping or many just outside the park
Canyonlands - can easily do Arches and Canyonlands from the Moab area or there Dead Horse State Park at Canyonlands itself - gorgeous views
Canyonlands - Needles area - dry camping in the park
Monument Valley - Gouldings RV for full hookups is the only only close
Have a good trip! Sept/Oct are ideal months.