I’m a big fan of the Shenandoah campgrounds, and particularly Mathews Arm (I was last there mid-June). The Big Meadows campground (I was last there last weekend) has more amenities in the campground (showers, laundry, wood, ice) plus a large campstore a mile away, while Mathews Arm requires a 2mi trip to Elkwallow wayside for wood, ice, and other campstore things. However, Mathews Arm is smaller/less crowded/typically quieter. From hiking standpoint, there are plenty of trailheads nearby both campgrounds.
Both take us about 2.75 hours to get to from the center of DC, leaving in the middle of the day. If you’re able, I recommend taking off on a Friday and getting in mid-afternoon for maximum enjoyment (less traffic/quicker drive, wood will be plentiful, you’ll put your tent up in the daylight, etc). This is especially true if you’re looking to go this weekend, as you won’t be able to reserve a spot via recreation.gov (too close to the day) and will have to do walk-in registration — the earlier you get there, the better!