The section that stands out to me is between Mesa Verde and Zion. I would head from Mesa Verde to Las Vegas and then to Zion, the North rim and Bryce, it will save you a lot of miles and will take you to hoover dam rather than having to detour or back track.

I doubt you will get a site in the Watchman campground in Zion and even less chance of a Riverside site, the sites that are better spaced as Utahtea was referring to. The resort mentioned "10 miles away" is called the Virgin river resort.

The only campground on the Las Vegas strip is Circus Circus and it is a parking lot with hook ups! Oasis is a resort but you will either have to walk 1/2 a mile to the bus stop, take a taxi to the strip or if you have a tow vehicle, you will be fine.
I would recommend the Oasis. They also run a free shuttle bus to the start of the strip where you can get the trolley rides along it. The shuttle does go around the houses a bit though picking up and dropping off at other hotels and a cab is most convenient.