I am not aware of a good long loop in the whites, so I think you should plan to use the shuttle. I also avoid hiking in the whites in Aug, because as beautiful as they can be, they are overrun with hikers, particularly on the weekends. You are going to have give up any notion of avoiding crowds and consider planning around crowds, including finding places to camp, one of your major obstacles. Lincoln and Lafayette out of Franconia, and Mt Wash and that whole surrounding area get a lot of foot traffic. But don't let this dissuade you either, the hike will be well worth it.
I don't have my maps out, so I can't tell how far it is reasonable to go in 7 days, but you will be following the AT for most of the way. You certainly want to do the north presidentials, Jefferson, Adams and Monroe for sure, and so dropping down to a bus station there near Gorham is an obvious end point. Going past that point up over wildcat takes you into Maine, so probably farther than you want to go. Whether you start in Franconia notch or further "south" on the AT, I can't tell. If you want to get off the AT and away from the masses, you could do spurs off into the Pemi or into Dry River wilderness, but any time you come off the ridge of the AT you are adding a lot of up and down.
Good luck and enjoy the trip.