For more info see file:///C:/Users/jon/Downloads/2023_IT-511_Individual_Income_Tax_Booklet.pdf
page 24 (pdf page 25).
re: withholding and overpayments. Remember withholding is only an estimate of you taxes. It is usually wrong one way or the other. The goal is #1 to make sure you have the cash to pay when filing your return if it were too low, #2 to get as small a refund as possible (because that is an interest-free loan to the state), and #3 make sure that there is no penalty/interest for not paying enough during the year.
After you do your first year of this, you will see whether the amount actually owed (not withheld) to SC is large enough to require withholding or estimated tax payments. If not don't worry about it. If so get the right amount withheld or make estimated payments. For more info on the SC rules applying to non-resident withholding you might start with https://dor.sc.gov/resources-site/lawandpolicy/Advisory%20Opinions/RR22-3.pdf
page 5-6 and examples 3 and seem relevant
I think your employer should only be withholding to SC on the part of your salary that was earned while in SC and 100% of your salary to GA.
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