I haven't fished Fulton County--- maybe others will weigh in. In the mean time, here’s a link to NY’s DEC Region 5 Fishing page. (Region 5 is considered NY DEC's "Adirodack/Lake Champlain Region")
Adirondack/Lake Champlain Fishing - NYS Dept. of Environmental Conservation
If you follow the “fishing spots” link for Fulton County you’ll see some waters for Brown Trout are listed including Canada Lake and Cayadutta, Meco and Timmerman Creeks but i haven't fished any of them. Not sure how those creeks would be in the heat of the summer for trout. If they are spring fed they could be fine, otherwise high water temps (water temps of 69+ degrees) might be a problem for trout survival, but they may fish well for other stuff (bass) even if there aren't many trout.
You’d also be reasonably close to some better known fisheries like the NY stretch of the Batten Kill in Washington County, Schroon River in Warren County as well as within striking distance of many Essex waters to the north like the West Branch of the Ausable and the Boquet River. These are all also in Region 5 and any of them would be worth exploring if you can travel a bit.
To the west in Region 6 (considered NY DEC's "North Central NY Region") you also have West Canada Creek in Hamilton County and Oriskany Creek in Oneida County they should not be too far from Fulton county.
I've fished most of these mentioned above, as well as a few lesser known streams in these areas.