Yes, I have 'done the dragon' many times, as it is only about a three hour trip from here. It is a very challenging road, but it is only 11 miles after all, so don't come all the way from Maine just for the dragon. But there is some great riding along the 36-mile Cherohala Skyway also, or you can loop through the Smokey Mountain National Park for a nice all-day trip. Maybe after one day in the park and another on the skyway/ dragon, plan on spending a couple days on the 470-miles of the Blue Ridge Parkway. The views are great, the roads are fabulous. But there is a lot of tourist traffic, and some stretches of the parkway are closed due to storm damage and mud slides from the trees beoing blown down this summer in a couple hurricanes.
As for the dragon, avoid it on weekends, as it is wall to wall bikes and sportcars. We usually go on a Thursday. Go at your own pace, and you will be fine. If someone comews up behind you, just let them by. The Cherohala Skyway can get you in more trouble, as people expect the dragon to be tight, but the sweeping turns of the skyway temp you to run faster than you should, then a super-tight curve will sneak up and get you killed.
You might want to stay in Asheville, NC (I can give you hotel info if you want). Take the Blue Ridge Parkway down from Virginia, stay in Asheville, do a day trip to the Dragon and the skyway, the go up Devil's Drop and run the Blue Ridge Parkway over to Cherokee as a day trip, and then spend a day in the smokey mountains national park.