http://www.dec.ny.gov/63.html
Watertown is on the western edge of the Adirondack Mountains. The Adirondacks are a state park of about 6 million acres (not all of it is state land), mostly wooded. There are about 1000 bodies of water in the Adirondacks, if you count ponds, lakes and streams (in the Adirondacks, anything less than 2 miles long is called a pond). The vast majority of the Adirondacks is open to fishing and hunting. What land is posted is usually posted by a hunting club that leases land from a timber company.
Deer and black bears would be the big game available. Small and upland game opportunities abound in that area. Generally, small game season is September through February. Big game is October- December. Turkey season in the spring is May.
Ft Drum itself is a huge area, some of which is open to hunting.To the west of Watertown is Lake Ontario, which is one of the best salmon fisheries in the US.
Check with the people at the fort- they'll know what you have to do to qualify for a license- or if you even need one. I seem to recall something in the regs about military personnel being able to hunt without buying a license.
The short answer is, if you want to hunt, fish, or camp, you've hit the mother lode- it's a great place for any of those activities.
The bad news is, winters are tough there.