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.
What do you mean by they will give you a credit? Will I owe South Carolina state by not having paid taxes to them through my income?
Possibly. If you did not pay enough taxes for your income then you will owe SC. As @SteamTrain mentions SC will give you a credit to mitigate the SC taxes on income that was taxed by NC.
When you complete your non-resident NC return be sure to take note of the income that was taxed by NC and the tax on that income. You will use those numbers to complete your SC resident return.
In TurboTax, be sure to complete your non-resident NC return before you do your home state SC return, so that the program can calculate and apply your credit correctly.