We were there March 25-April 1 and I was prepared for the worst! We made sure we were there for every park opening and EMH we could do and grabbed fast passes as soon as we could (but saved them for later in the day). The parks were actually quite pleasant from opening until about 11 am...then they got CRAZY. Wall to Wall people! So...we walked in and out of lines early in the morning and went back to our hotel in the afternoon. We'd come back early evening, use our fastpasses and catch up on evening shows, shopping, etc.
It really wasn't too bad, we'd definitely do it again. I've never been in June...so I can't help you there!