With two kids in school, we too need to go when they are on vacation (I know some people do take their kids out of school. we tried it once, but it really didn't work out to well). We have gone during different periods in the summer, Thanksgiving and Christmas. Early June was the best.
During early June some parts of the country were still in school. It's not quite as hot as July-August, but plenty hot for swimming, getting wet (which is a must for our kids). The crowds are there, but they are manageable.
The absolute worst I have seen was Christmas Day 2003. Talk about wall-to-wall people. I will never go Christmas week again. Thanksgiving was much better then Christmas (at least right up to Thanksgiving Day), plus it is warmer than at Christmas.