If you are really set on going there in June you can. I have been out there the first week and a half of June, it was hot but doable. You will just have to chase the shade. 90+ degree in the sun out there is way hot.
First couple of weeks of June shouldn't be too bad, not much for humidity so that won't be a problem. The creek is going to be 5-10 degrees cooler then Moab since the creek is at a higher elevation (right around 5500) Monticello is slightly higher then the creek. That is why you are seeing the difference in temps. Locals climb year round in the shade, just depends on how you handle the heat. You will get a afternoon thunderstorm once in a while but the rock dries super quick.
I would get up and be at the crag just as the sun hits it, chase the shade and maybe take a long break towards the heat of the day. Then hit it hard towards evening. Nights will be comfy for sleeping not long after the sun goes down. I have been going there for the last 7 years, I've spent a month or more each year. Have fun, pm me if you want more info.
If I had a choice, I'd go in mid Sept.- mid Oct. its perfect!
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