Elkmont is the most isolated big campground I've been to in this part of the US. Watch out if you're taking the Little River road from Townsend in a big RV: it's really twisty and narrow with rock overhangs. I felt like my 28-footer barely made it. Don't know if a big rig would fit. Elkmont is the antithesis of nearby Gatlinburg which is very ticky-tack. It's probably the most peaceful spot I've ever taken the RV. Tall forest canopy covers the campground: it's all shade. The streams in the campground can't be duplicated elsewhere. My kids love this campground but they're outdoor types. If your kids need TV, Wi-Fi, cell phones, game rooms, etc., please take 'em somewhere else less quiet.
This Boone KOA is our favorite place for "kid" camping. The climate is wonderful: 70s in mid-summer at 4000 ft elevation. Pool and game room are much appreciated by the kids: the pool is nice and clean. They have what would be a very nice little putt-putt if some work was done on it. As for the sites: I've seen some reviews state they were trailer park-ish and I guess I'd have to agree. But the nice thing about this KOA is the other visitors (usually from the GA/TN/NC areas) are really friendly and it's a pleasure to be in close quarters at Boone. Staff is very friendly and sociable as well. The other infrastructure (hookups, access, etc) is just fine. If you're a no-kid couple looking for isolation or driving some giant big rig this might not be the place, but for family fun with a pack of kids Boone KOA can't be beat.
Big Meadows is the flagship operation for Shenandoah National Park, and it merits repeated visits. It has the best set of ranger and wildlife programs in this region. The climate is good at 3600 ft elevation. There are a wide variety of sites depending on loop. Our family customarily likes to use Big Meadows as the last night or two of an extended trip as it's not far from our home. There's little to no leveling or site preparation to be done and lots of facilities and events to entertain the kids; it makes for a nice low-key diversion to end a journey. If coming in October, I'd advise a midweek trip to avoid the tourist traffic pouring out of Washington, DC. Even then, Monday through Thursday are nice and quiet.
Peaks of Otter is an outstanding campground for families with children. It's an NPS campground of the old CCC type, as are all the other Blue Ridge Parkway federal facilities. These facilities were originally designed for car camping back in the 30s, and indeed loops A and B are very nice for this purpose. Loop T was adapted for RV use. There are about 50 sites in loop T. My motorhome is 28 feet and could fit in almost any of the spots. I can think of maybe ten sites in the loop which could accommodate a diesel pusher. Being an NPS facility: there are of course no games, pool, etc., but for ranger programs, exhibits (i.e. the nature center and Johnson Farm), kid-friendly hikes, fishing, and all the other things NPS is known for, it's more than adequate. The camp bus operates to the top of Sharp Top mountain and it's well worth the price of admission. Peaks seems fairly lightly traveled. It's south of the Shenandoah NP crowds and I have never seen the campground full except during mid-October. Strange, because I have traveled the Blue Ridge Parkway many, many times by RV and consider it the best in-park facility.