Residents and part-year residents are the only ones who can claim a credit for taxes paid to other states. You can't claim an other state credit on both a resident and non-resident return. You'd be double dipping.
New York does tax telecommuters and if your company continues to withheld New York state tax, it means they still consider you to be a NY-based employee even though you permanently moved to South Carolina.
You are correct about New York City tax. Are can claim a refund for NYC income tax for the period you lived in SC.
If your only income was from your NY job, you probably will owe some money to SC. As you said, the SC rate is higher than NY. It's 7% on incomes above $15,400 while NY doesn't go past the 6%+ until you earn more than $1,077,551.
You would NOT be eligible for the credit for taxes paid to other states on Form IT-203. You are seeing that credit because you moved but continued to earn NY source income while living in SC.
That portion is the part that would be covered by the credit on your SC return. You should delete Form IT-112-R. During the period you lived in NY, you only paid taxes to NY and NYC so there is no credit for that portion.
Check South Carolina. Make sure you are filing a part-year return and that your income allocations are correct. It more generally more advantageous to use "Nonresident OR part-year resident filing as a resident" than "Resident OR part-year resident filing as a resident." You can toggle both ways and check the tax due box.
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