If I sell my second home in SC, do I need to pay capital gains tax if I purchase another second home in SC with the profit made?

Yes.   What you do with the proceeds of selling that home is irrelevant.   The law that allowed you to reinvest in another home to avoid capital gains ended over twenty years ago.

 

SALE OF HOUSE

 

If your gain was more than  $250,000 filing Single, or more than $500,000 filing Married Filing Jointly the sale must be reported on your tax return.  Whether you re-invested the gain in to another house is irrelevant.  If you  have a Form 1099-S go to Federal>Wages and Income>Less Common Income>Sale of Home (gain or loss)

If you owned and lived in the home as your primary residence for at least 2 of the last 5 years on the date of the sale, you do not have to report the home sale if the gain is less than $250K filing Single, or less than $500K filing Married Filing Jointly (and you both owned and lived in the home for at least 2 years).

• If you are using online TT, you need Premier or Self-Employed software to report the 1099-S
• .https://ttlc.intuit.com/questions/1901135-i-sold-my-home-at-a-loss-is-this-deductible

Yes. The two homes have nothing to do with each other.