This may be a little late for your trip as it is almost mid September but I was surprised that Ledgeview CG did not come up. We stayed there last year and it seems to meet all your requirements. It is moderately priced, all large wooded sites, a campground store for necessities and a small pool. Good jump off point that won't break your wallet which is just a few miles from the village. I have not stayed at Schroon so I can't give a comparison, but either one will probably meet your needs so have a nice trip.