Wellesley Island state park up near the Canadian border is open all year with elec.
It is at the North end of 81.
I am a summer resident of the Thousand Islands area and have made the trip to Manorville, NY for years.
495 Throgs neck GWB to 80W to 380 to 81 nice roads toll free after GWB except for $1.00 at the Delaware water gap and the toll over the thousand Island bridge.
Your New York fishing license means you can ice fish the St. Lawrence.
Towns nearby include Clayton, Alexandria Bay and of course 25 miles away is Watertown, NY
Have an enhanced drivers license or passport then Canada is 10 minutes away.
Towns to explore there include Gananoque (1/2 HR along the river) and Kingston about 1 hr home of the international Hockey hall of fame. There is a casino in Canada near Gan as well. Ottawa is about 1.5 hours away