Do NOT go to Dallas! It would drive you brits crazy as you must have a car to get around and there isn't squat in the way of tourism.
My advice: Washington DC, New York, Boston, and Kennebunkport, Maine. The US east coast is very scenic.
Washington DC is incredibly beautiful and has features a lot to do. I highly recommend visiting the Smithsonian and maybe stopping alongside the Lincoln Memorial. A little visit to Capitol Hill might be nice. You could probably do DC in 2 days if you wanted to(fly in late afternoon, have to visiting days, fly out late the second day).
New York has a bit of everything. I recommend Times Square, a visit to the top of the Empire State Building, a walk through Central Park and a visit to Museum of Modern Art. The top of the Empire State building has some of the best views of Manhattan. There's a big tourist thing there.
Boston is like New York, but more English and less cluttered. Walk the Freedom Trail which is a 2 mile walk featuring the places where you all lost to us
. It does seriously though, show you a lot of the finer points of interest in the city. The Boston Duck tour(although a bit childish) takes you several places and it doesn't take all day either. It maybe takes about an hour. Boston has a lot of good restaurants. I highly recommend visiting the Charles River.
Maine is much more tucked away, but is maybe a 1.5-2 hour drive from Boston. Kennebunkport is somewhat of a tourist area and features a lot of one-of-a-kind type shops. It can be crowded, however. I recommend staying at the Beach House which is right on the water. Unfortunately, the main part of the town is farther inland although I think there is bus service from the sea to inland, if there is, take it as parking can be a pain. There is one small parking area and it costs an arm and a leg. I recommend taking a .5-full day trip to a big outlet area in Maine(unfortunately, the town has escaped my mind). There you can find LL
Bean's flagship store and just about every other designer you can find. Mable's in Kennebunkport has some great Clam Chowder and Clam Shack has great fried shrimp and clam strips.
If there is any place to eat sea food on the east coast, it has to be Maine.
I recommend visiting Hawaii, but that seems like you parents do not want that. Hawaii, being such a tourist location, features unbelievably high airfares and hotel prices.
Give me a break, I created this username when I was a kid...