Work in SC, but taxes mistakenly withheld for NC

I live and work remotely in SC but my state income tax for 2021 was withheld for NC erroneously. My employer states that this can not be corrected on their end. I had already filed my federal return before I realized this error. Should I file for NC as a non-resident and report no income to recover the withholdings, then file an SC return? Do I need to file an amended federal return?