Q. Do I need to claim NC taxes on my wages?
A. Simple answer: Yes, you should have both SC and NC tax withheld from your wages, if your employer is willing to do that, for you.
This is the general rule: The income is work state (SC) source income since it was earned there. Resident States (NC) tax all their resident's income, regardless of where earned. You will file a non-resident tax return for SC and report the SC income. You will file a full year resident return for NC, reporting all your income. NC will give you a credit, or partial credit for any tax paid to SC.
Since, NC generally has a higher tax rate, you will probably owe NC (it depends on your income, there's a point where the SC rate exceeds NC), you should probably have some NC tax withheld.
But it can get complicated, you may just want to "wait and see" for the first year (there may be a small underpayment penalty to NC). Paying quarterly taxes to NC is another option.