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.
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.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.