First, since New Hampshire, as you note, does not tax wages, you will not need to file there.
Next with NJ as your resident state you will file a resident return there for all income and all sources. Since nothing was withheld for NJ you will owe taxes there.
Finally, SC law states employers will not change withholding due to a worker temporarily working from home outside of SC. Further SC will not change it's taxing of workers who were normally working in SC but temporarily now work outside of SC from home due to the pandemic. Here is the SC tax letter and then an extension letter through Dec 2021.
What that means for you, assuming you did not physically work in SC pre-pandemic, you would not need to pay SC taxes. However since they withheld taxes, you will need to file a non-resident return there to get a refund of those taxes.
It appears SC should not be withholding, but NJ should. You would need to change that with your employer. In the future you would only need to file in NJ as a resident there.
To generate the SC non-resident return go to "My Info" and scroll down to the "Other State Income" topic. Indicate you earned income in another state and select SC.
NOTE: You will not be eligible to claim a credit for taxes paid to another state on your NJ taxes since you will not be paying any taxes to another state (getting full refund from SC).
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