Take a look at South Carolina and its brilliant beaches. Also try to include Charleston and Savannah. We went when we only had a two year old.. now 16.. but worth checking out nevertheless. Not sure of its theme park status though.
Florida keys was perfect for us with 3 dc's aged 10 - 14. Loads to do on the water, trips out to Miami, the everglades and busch gardens. temp never gets above 100. our 3 want to go back there rather than the theme parks which says a lot. if you are outdoors types theres cycling boat hire, snorkelling at john pennekamp park, canoe trips, and masses to do.
We're planning a 2 week holiday next may for my 11y dd and 8ds.they want to go back to Florida (already been twice)but I would like to try somewhere different. Ideally it would include a theme park,beaches,decent shops and food. We love USA so happy to go back there but would like to try somewhere different. We've done the LA to SF road trip when the eldest was a toddler(not keen on LA)but haven't been to San Diego or beaches (palm springs,Miami ) Any suggestions?