We are from Tennessee (and I am from Alabama), so we know all about Southern heat and humidity! We went to Disneyland for the first time June 22-26, 2013. It was wonderful!! The mornings actually started out cool. It was overcast, and some mornings we wore light jackets or long sleeved t-shirts. The parks open early. If you stay onsite, your early entrance is at 7:00! We would get to the park at 7:00 and stay until about 2:00. We'd usually eat lunch somewhere in the parks. By then, the clouds had burned off, the sky was blue, and the sun was intense. It's hot, but not humid. We would come back to the hotel and swim/nap until late afternoon. We stayed concierge at the Disneyland Hotel (a big splurge for us!), so we would usually freshen up, pop up to the concierge lounge for hot appetizers around 5-ish, and then head back to the park until closing. It got cool again once the sun set.
I would not hesitate to go back in June! Enjoy planning your trip!