For Moab, make sure you check out my long post on most things Moab:
A Primer on MoabMaybe take one day from the Grand Canyon and stay around Flagstaff for a night. Several interesting NM's in the area: Wupatki, Walnut Canyon, Sunset Crater. Flagstaff is a little higher and thus a little cooler. Unless you're doing some hiking, you can see much of the south rim in a full day. In late June, it will be blazingly hot much below the rim.
In Zion, hike trails on the east side of the canyon in the a.m. and the west side trails in the p.m. That way you'll be in the shade of the canyon walls for much of the hike. Pick the warmest day and hike up the Zion Narrows. Bring water shoes or rubber sandals as you'll be wading for much of the hike. At least 10 degrees cooler in the Narrows so a great place to spend a hot part of the day.