Designated campsites otherwise generally probably further of a hike in than you want (e.g. Giant's Washbowl near Chapel Pond is 3/4 mile and 500' uphill....)
We had an adventure with one of these campgrounds. Had a hard time figuring out if we were car camping or winter backpacking. Fun, but a bit of a hassle.
The best news is the hostel in Keene Valley, right across from the diner. $20 a night, clean, good showers, a full kitchen, run by nice people.