The only Shenandoah campground that will be open is Big Meadows (until November 17); there are no electrical hook ups. There are generator and non-generator sections, so choose your site wisely. I would recommend just getting one when you get there; it shouldn't be too busy if it is cold.
If you don't abuse your battery, your generator should be able to top it off for the overnight. At BM, you cannot run your generator after 9PM during late fall but if you don't run every light you should easily make it until 8AM when you can use the generator again.
Typically, when we stay there late fall, we can go for three days without using the generator so I don't think you would have a problem.
Don't forget to add water before you park, when it is convenient. Not every site has access to a water spigot and your rental may or may not include a hose.
Enjoy your trip! We love SNP and camp there at least three times a year. If you have any questions, pm me...
2015 Jeep Willys Wrangler
2014 Fleetwood Bounder 33C
States camped: all but Hawaii
more than 1700 days on the road