There a couple of Trails Illustrated Maps from Nat Geo that well describes the dirt road systems in the Dak’s. There’s a digital version that might be useful.

In my research of the areas NW of Saranac Lake, west of Tupper Lake, east of Long Lake and the Moose River Plains, with an eye towards bikeable dirt roads, I noticed there are a lot of road side primitive campsites scattered in these areas. Fire ring, privy, picnic table is typical. Some of these roads are thru routes, so possible to plan a trip with dirt roads as part of the route. All primitive sites are first come, first serve, so possibly a weekend might be busy.

It would be great if the DEC allowed off-season camping at the closed public campgrounds, as does Vermont. Knowing NY State, they likely do not, but a call will tell you.