I ran Skyline Drive and the Blue Ridge Parkway South in late September last year and had no trouble at all with a 34 foot Class A hitting every single campground along the way. Most of those campgrounds were designed for 1950s Woody Ford station wagons and not modern RVs - - not that they would be open this time of year. And I'd be watching the weather too.
If you do a little map work you should be able to identify major roads crossing the parkway where you could drop down into the valleys for Walmarts and fuel etc. Don't just arbitrarily take a side road down off the mountains because you may end up following your own tail lights.
Somewhere roughly around milepost 300 or so there is an excellent Walmart at the little mountain town of Spruce Pine North Carolina. A highly recommended stop.
Through some of the tunnels you may feel a little more comfortable straddling the white line to be away from the arched side which I did assuming there was nothing coming towards me of course. But I don't know that it would have really been necessary.
If you do a little homework with a map and pre Identify some Walmart's and such off the parkway I don't think you'll have any trouble at all.
