We've stayed at Lone Rock on Lake Powell. You can drive down to the water but be careful of soft areas. We like to stay on top of the bluff on the left just after passing the ranger contact station. You'll get beautiful views of the lake and you can easily walk down to the water.
Camping - Glen Canyon National Recreation Area (U.S. National Park Service)
Agreed....it starts getting hot in May.
Where else are you planning? The southern Utah national parks are a must and they're all in that area - Zion, Bryce, Capitol Reef. Each national park has a web site similar to the above link for Glen Canyon. They give lots of great information. Check them out.
The North Rim of the Grand Canyon is also nearby and it's a much cooler area as is Bryce. DeMotte national forest campground is very nice for the North Rim. You will also find a very good flat gravel national forest road just south of DeMotte (right next door) and heading west on it. Drive down about 1/2 mile and you'll start seeing wonderful dispersed camping sites. Both places are at the North Rim park entrance area.
Also consider Arches Nat'l Park and Canyonlands. Lots of boondocking in the area. Here's some BLM camp sites:
Have a great trip from another Michigander.