The outer banks of NC are great. Very few tourists, beautiful beaches, and when the kids get bored you can take them up to Kitty Hawk and fly kites all day long. There can be some decent waves in the Atlantic, which are a blast for kids.
The gulf is also gorgeous. Depending on the time of year and weather, you can have crystal-clear blue water that's perfect for snorkeling or sea kayaking. Around the panhandle of FL, there's also pristine white sand. A lot of places are heavily developed, though, and you'll be dodging tourists, but there are still plenty of campgrounds and hotels off the beaten path.
The smoky mountains also come to mind. You could drive down the Blue Ridge Parkway while the leaves still have the fall colors. There are lots of waterfalls and short easy hikes that are good for kids. This time of year is actually prime time to see the leaves in their full splendor. I'm betting around late october / early november will be the peak.
The cumberland mountains are similarly gorgeous. You could hike around Fiery Gizzard state park in a day, then hop over to Nashville for some culture.
This is also a fine time of year to go whitewater rafting. It's not that cold yet, and a lot of rivers are running. The downside is that a lot of outfitters will have already closed for the season, so your options may be limited.