One consideration I would have about some other areas of NY are the rates at which people are leaving the state. The town we moved from had a 22% population decrease from 2010 to 2018. And 37% since 1990. I don't even think teaching jobs were secure because they were trying to merge with other districts while we were still there. If you do need/want to move, it's quite difficult selling a house where the population has dropped so drastically that there aren't any people looking to buy.
We also live in a rural area near Winston-Salem, NC. We picked this area by doing research online and then making a trip to drive around and check out the places we had on our list. I would highly recommend that to the OP. Some places may sound great (especially with school ratings), but when we saw them in person the traffic or the feel of the area was a very quick "no" for us (Charlotte suburbs).
Yes, people from NYC tend to refer to the entire state as "Upstate NY". I always thought it was funny when my NYC family would refer to an area as "upstate NY" (less than an hour out of the city) and my Central NY family would refer to the same place as "downstate".