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I have been here less than a year but I would say Cochise would be too hot (forecast will be in the 90s in several days).  I think Sedona and northward near Flagstaff would be a better trip.

I would suggest Paradise Forks or Volunteer Canyon near Wiliams, AZ. These are not good for long routes, but are beautiful crack climbing on basalt in a remote forested setting. 

If you're set on going south, you can certainly make Babo or Cochise work with a very early start. I did Babo on a day when the forecast was low 90s in Tucson, started hiking at first light, and with a breeze I was actually shivering at the first belay. It was boiling hot hiking back down into the desert in the afternoon, but at least you'll have cold beers/ac awaiting you soon at that point. 

Another good option for you is Reef of Rocks on Mt. Lemmon. Good adventurous climbing on Granite domes up to 300 feet high, and will be good temps in early June. 

I second Reef of Rocks on Lemmon or Rap Rock on Lemmon as both will be cool.  Babo could work.  Do not leave anything in your vehicle at Babo, just to be safe.  However, if you've never been to Babo before, finding the Southeast Arete will be a major challenge.  Also, the decent off of Babo is a real route-finding adventure.  If you do Babo, find someone who has been there before.  I've done Babo around five times.  Definitely a sacred place.

Check out the stuff in Oak Creek Canyon for trad multipitch.  Especially look at Sterling Pass and West Fork of Oak Creek. You should be able find shade depending on time of day.  Also certain routes in Sedona would be in the shade.  For exmaple "oski," would be super warm approach but the route would likely be in the shade.  Also climbing in the evening in Sedona is tolerable this time of year, especially on climbs like the Mace where some of the pitches are shaded.  

Tradster wrote:

I second Reef of Rocks on Lemmon or Rap Rock on Lemmon as both will be cool.  Babo could work.  Do not leave anything in your vehicle at Babo, just to be safe.  However, if you've never been to Babo before, finding the Southeast Arete will be a major challenge.  Also, the decent off of Babo is a real route-finding adventure.  If you do Babo, find someone who has been there before.  I've done Babo around five times.  Definitely a sacred place.

FYI Rap rock on Mt Lemmon is closed for falcons until June 30.