June is right in the verge - it can be cool or hot. We spent about a week in that area last June, and some days got hot - high 90's, but others stayed cooler. We were there in the middle of the month, and it does get hotter as you get closer to July.
It's a bit of a gamble (weather-wise), but we took that bet because it's such a beautiful area.
2003 Country Coach Intrigue, Cummins ISL 400
Toad: 2006 Jeep Wrangler Unlimited (LJ) toad, with just a few mods...
Other rig: 2005 Chevy Silverado 3500 Duramax Dually / Next Level 38CK Fifth-wheel Toy Hauler w/ quads, sand rail, etc...
We were there last June the weekend following the cool weather and it was 102. We cut our stay short and headed further North. Great area tho!
Also a big change was the bugs came out in force the hot weekend. We had friends that were there the weekend before and there wasn't any bugs. Take a good High count Deet bug spray with you. As posted earlier, it can be freezing or frying!