Here are a couple of links to the State Forest camping area where I have spent most of the last three summers, including this one:
https://www.dec.ny.gov/lands/7999.html and
http://andyarthur.org/brasher-falls-state-forest.html
The second one is a little out of date in that the water source has been permanently capped and some of the sites no longer have usable outhouses and/or firepits or picnic tables, but it is never full so you should be able to find a site that meets your needs.
I haven't visited the Moose River Plains area mentioned in the second link, but have heard great things about it from other campers here.
Another area worth checking out on the DEC web site or Andy Arthur is Mountain Pond near Paul Smith's in the Lake Placid/Saranac Lake.