Here are a few more thoughts concerning the Skyline Drive (SD), the Blue Ridge Parkway (BRP), Shenandoah National Park (SNP), and Great Smokey Mountains National Park (GSMNP):
From the northern (closest to Baltimore) end of the SD, it's 575 miles down to the end of the BRP. That includes 105 miles of the SD and 470 miles of the BRP.
The SD is the main access and travel route within SNP, essentially bisecting it from north to south. The SNP is long and narrow, atop the Blue Ridge Mountains, and the SD runs along its spine for its whole length.
The BRP then connects SNP to the GSMNP, some 470 miles farther south. The BRP is its own National Park unit, albeit a long (470 miles) and very narrow in width, varying from < 100 yards to just a mile or two. The stated purpose of the BRP when planned in the 1930s was to connect SNP and GSMNP.
Travel is slow on both the SD and the BRP. Nowhere are speed limits > 45 mph and many segments are but 35 mph speed limits. Particularly on weekends, traffic will be rather heavy, especially close to cities. Once you get south of Roanoke, VA on the BRP, traffic will thin out a bit until you get back into it in the Boone/Blowing Rock, NC area. Allowing time for some of the hundreds of vista points and exhibits, it's quite likely you would average but 25 mph over a "beginning to end of day" basis. That said, a leisurely trip down the parkways, from end to end, is at least a 3 to 4 day endeavor. If you stop a lot and/or get off the routes to visit Charlottesville, Roanoke, Blowing Rock, Asheville, and the like, it's more like a 5 or 6 day trip.
Some think a trip down the entire length of the SD and BRP is "too much of a good thing", and some people even experience a bit of motion sickness due to the effects of continuous curves and hills for days on end. I mention this just so you'll use some of your planning time exploring the many opportunities to get off the SD and BRP via US 29, I-81, I-77, US 421, and a variety of roads in the Asheville, NC area.
From her many posts, I am aware of Lifemagician's frustration with the weather in the Blue Ridge Mountains of VA and NC. It is most certainly possible to have heavy fog (clouds, actually) for parts of several days, and sometimes for days on end. That's where your Plan B comes into play, as normally just coming down off of the higher elevations into either the Shenandoah Valley or the VA or NC Piedmont gets you below the clouds.
Have a great time planning and taking your RoadTrip!