We'll be there at the same time - we're going June 15 - June 21!
Since early morning is not an option for you, I would suggest you consider doing an e-ride night. There is one scheduled for Tuesday, June 17 and another for Friday, June 20.
June can be really hot, so plan lots of breaks in your day. We always hit the parks very early then leave to swim for the hot part of the day and go back to the parks in the evening. I'm sorry I can't give better advice for someone who isn't a morning person, but maybe someone else here can help.
I would definitely recommend looking at the June hours at Disney's site so that you can make the most of the evening hours.