New York follows the "convenience of the employer" doctrine. This means that if you're working remotely for the company's convenience, rather than your own, then your income is not subject to New York income tax. Since you were hired specifically to work remotely, IMO your income is subject only to SC tax, and your payroll department is withholding correctly. Tip: be sure to keep that offer letter with your records.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.