You’re a little under and hour from the Black River in Watertown (not the Hwy 7 one). Its a great river for both playboating and river running, it is ice free and has water almost year round and has features at every level.
Check it out on americanwhitewater.org and while you;re there look at the rest of NY state. Lots of choices.
I live in Kingston and have been paddling for the past few years. Spring is a great time to be a paddler in Kingston, after that it certainly dries up. The Ottawa River is about 2.5 hours away, the Gull is 3 hours, the Gatineau is about 3 hours. The Watertown Black is about 1.5 hours (including border crossing). The Watertown Black is a great river for summer paddling (River running as well as park-n-play) but make sure you go with someone you know. During the summer it is not that popular a river and so there is often nobody there, which is great if you are a skilled paddler but terrible if you are still learning (like me)!
PM me if you are looking for someone to paddle with.
If you don’t mind about an hour and a half drive from Ktown and can cross the border past Gan, you could try the Black in downtown Watertown if you’re looking for park and play. There’s also a run going (I believe) out of town on the Black, but I’ve not run it and I think there’s some more demanding paddling involved in spots. Maybe somebody else could help out with info on that one. The river level is usually on the Boaterboard What’s Running page updated by Thom. Cheers.