It will be quite warm in Moab in mid-June. Although with the drought we're now having, with your ability to bring rain, we'd all like you to come out soon to get the precipitation going again. ;)
As noted, Bryce will be cooler as it is higher in elevation. One can plan hikes at Zion so you do hikes on the east side of the canyon in the a.m. and the west side of the canyon in the p.m. to take advantage of the shade. On the hottest day, hike the Narrows. I find most of the hikes at Capitol Reef to be pretty sunny. Not nearly as many hikes around Kodachrome although Calf Creek falls is worth visiting.
Maybe look for some higher elevation hikes in someplace like Mesa Verde NP or Black Canyon of the Gunnison HP? Personally I'm finding it more difficult every year to deal with really hot temps.