Kapersky,
I would guess that this would be the same as working in a state in which you do not reside and having to "pay" taxes in both states. Well, this is how I think this would work in your case, although you should confer with a CPA or a certified tax preparer. Let's say you won 5K in SC, you would get a 1099 from SC indicating your winnings. You would file your NC taxes, with a SC tax form in which you would indicate your miscellaneous income as lottery winnings, pay your taxes on it, and list this as taxes paid to SC on your NC return, include a copy with your NC return, where you would write it off as taxes paid to another state. But, I think you've probably wagered enough to claim "that you've also lost at least 5K", in which case you would end up in actuality not paying any taxes on your 5K of winnings.
Hope that makes some sense. If not, as I said, confer with a CPA. That's what I do when I win big!
'chaser
It's unlikely though that they'll come after you for the taxes if you pay NC and the Feds.