I'd maybe try Smokemont. For you, the Bradley Fork and Oconaluftee are right there, and you can be in Cherokee in just a few minutes to entertain the family.
Since Smokemont takes reservations, you wouldn't have to worry about getting there and not getting a campsite. The last time I was at Cataloochee - August of last year while school was still out - we got there on a Thursday, and I think it was slap full by that night. It definitely was by Friday night.
JSouth's link will get you to the Smokemont reservation page.