I left the Texas Hill Country in early April, and the temps were already getting into the 80's during the day. Today, I'm in Rapid City, SD. It's May 2nd and it's suppose to hit 88 today!
Point is, you'll run into weather extremes wherever you go; regardless of when you go. That being said, as a general rule, it's going to get hotter and more humid as the summer progresses. So, I'd be inclined to do the south earlier in the season, and finish in the north. JMHO!
Sounds like a great trip for the whole family, whichever way you do it! Enjoy!