Virginia may tax your income. South Carolina may also. The provision you refer to are when two states have a recirpocal agreement, which does not exist between Virginia and South Carolina. The normal arrangement is that your resident state (South Carolina) may tax all of your income no matter where you earn it, and your nonresident state (VA) may tax any income earned there.
However, to reduce the effect of double-taxation, your resident state will allow you to claim a credit for the tax you pay to Virginia. Prepare the VA nonresident return first so that TurboTax can transfer this credit over to the SC resident return.
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