I'm curious, you're talking about planning around school and not wanting to take them out twice, but... aren't your kids still in school the first week of June? Most schools don't let out until the 2nd or 3rd week.
By the 2nd or 3rd week of June it'll be 7/8 crowd levels. The last week of June leading and first of July leading up to July 4th will be 8/9. It's busy. But doable. My *guess* is spring break might be a bit less crowded than the end of June, depending on when your break is and how close to Easter it falls. The weather would also be a bit better, less hot/humid.
However, if you can actually indeed go the first week of June, then that would probably be less crowded than spring break.
Our school always gets out the Friday before Memorial Day and school returns like the last Wednesday of July. So the first week of June (I would not go Memorial Day week unless that would be advised but I would think not) would be our second week of summer break. Spring break the last few years has been the week before and the week after Easter. Fall break is the week before and week of Columbus Day.
This is typical of the schools in a couple counties around here but I know the city schools have a different schedule, getting out more like second week of June and returning in mid August. I wasn't sure which was more prevalent nation wide.
Great news that most schools aren't out until June!