Other than Austin, Dallas. Primarily because it now has LRT service from DFW into the centre of the city where you can visit a local history museum in the old city hall, the JFK Memorial museum in the old Schoolbook Depository building, and go for BBQ a block away in a funky restaurant district before heading back to the airport. And if you have more time, it's a short walk up to a fairly extensive contemporary art museum.
The RBJ Library and Museum in Austin is a good look at the '60s in America, but most of Austin is in the evening.