Many people find it helpful to camp a few days close to the manufacturer after pickup to work out any little bugs or questions. Lucky for you, South Carolina has a number of state parks near Lil Snoozy
. We've camped twice at Santee, but that's the opposite direction from home for you. There's also a state park north of Columbia (Sesquicentennial) and one on Lake Murray (Dreher Island SP), just west of Columbia. We've not stayed at either, but perhaps someone can chime in. The one on Lake Murray offers lakeview camping.
I've never towed with anything as small as your Outback, so I'll defer to those who have. That said, if you've never towed anything before, the hills of the Parkway (and even the I-26 into NC with the long stretch of 6 or 7 per cent grades up and down at Green River Gorge) might be a bit of a challenge for the first trip out. Beautiful, though, and highly recommended for scenery.
For your August trip, I can recommend Letchworth, as did another person earlier. We had a camping family reunion there a few years ago. It is a very large, very beautiful park. The Finger Lakes are beautiful, but we camp in a relative's driveway there. We've camped in both the Adirondacks and Catskills. Also lovely, and some great views in many of the campgrounds. We did the trip around the top of the Lakes on the TransCanada a few years back, from NY to Minnesota, and thoroughly enjoyed that. Our neighbors to the north are great hosts, and have some wonderful camping to offer in the Provincial Parks... hard to even pick a favorite spot on that trip!
Enjoy your new camper!