you can park by the visitor center and just walk up to the 'dome', you'll only see an observation tower. But then you can hike in any direction and stay at your reserved park shelter/campsite.
I'm assuming you might be able to park at the dome area, call first to confirm. I know there are/or were shelters just a couple of miles down the AT in either direction.
It might be the highest point in the park, but not in north carolina, that honor goes to Mount Mitchell.
I found some pix from 2008 when my family stopped at clingmans