There are three shelters off the AT, Kephart on Kephart Prong and Laurel Gap near Sterling Ridge and Mt LeConte too. Good luck getting away from people here in the Summer, they're everywhere. Its been my experience the out of the way campsites are the first ones to be shut down for bear activity as no one wants to trek back and monitor any problems. There are however some campsites that are tucked away that don't see many hikers like the Greenbriar area 31 on Injun Creek, 32 on Porters Creek and 33 on Old Settlers trail. If you come in June just suck it up and get reservations for 13 Sheep Pen Gap, the azaleas on Gregory are worth putting up with people.
You have nothing to fear but fear itself (and bears)