State tax filing

Yes...Absolutely you do.

You have to file a Non-resident NC tax return for just the NC wages....then after that is done, you file a SC Resident tax return which taxes everything.

.................SC wants to see that you actually paid taxes to NC, 

...........................and then, when  SC calculates a tax on ALL your income...from everywhere (this includes bank interest and dividends, everything else too).  Once SC calculates their tax, then they will allow you to take a credit for part or all  the taxes you had to pay to NC.

____________*Answers are correct to the best of my knowledge when posted, but should not be considered to be legal or official tax advice.*