If you don't even know what the climate is like, I would suggest doing a lot of independent research on the state before "deciding" to move (Google is your friend...)
City-Data's NC page is also a good place to start.
Your budget would be hard to work in the urban areas, but lots of NC is rural. However, the rural schools are probably not the best (you get what you pay for, in terms of cost of living correlating with schools).
Regarding climate and weather, we certainly don't have the winters like you do, though many find our summers oppressive with temps in the 90s and 90+% humidity for sometimes a month. Of course it varies from one part of NC to another just as is the case in NY, so you can't really say "The climate of NC is like X."
Areas near the coast often get effects from hurricanes, and we have the occasional tornado, but in general it's not a state where you need to worry for the sake of the weather. Researching NC climate will tell you more, but remember, it greatly depends on region.