The resorts profiles on this website lists the drive times from the major areas at the bottom of the page.
The drive to Snowshoe is highly dependent on the weather on that day. Sure, if it's snowing hard or if the roads up there are still snow covered from a recent storm, I wouldn't attempt the climb up to the top of the 'Shoe in a FWD car. The rest of the time you'd be fine in a Corvette with performance tires (actually, that would be preferable).
As others have said, the drive to Canaan Valley is much easier and doesn't offer the same mountain hill-climb challenge as the last 10 miles to the top of Snowshoe.
It's also important to note that, thanks to the upslope, it can be pounding away up at Snowshoe and in the Canaan Valley while it's sunny and dry in DC. So it's important to check what's going on before you leave.
In addition to Silver Creek opening this weekend, more basin slopes at Snowshoe should be opening up too.
Snowshoe may be a challenge to get to, but once you're there, there's plenty of open slopes to ski/board.