Withholding sent to wrong state

My elderly Mom had cashed out an annuity and had state taxes withheld. She lives in Georgia but the withholding was incorrectly for South Carolina.  They used my mailing address (SC) since I am her POA and not her residency address which is Georgia. I do her taxes for her.  I see postings that say an approach is to file a non-resident form for her in SC with $0 income and get a refund for the SC taxes withheld incorrectly.  Is that the  best approach and what is the sequence of filing. Do I file Federal, then file the non-resident with SC and then file the GA return. I don't have to wait for the SC refund before filing GA do I?  It seems I should be able to file both the SC non-resident and GA in parallel.  So, she will have to pay the GA taxes and then count on getting the money back from SC, true?

Or is it the annuity company responsibility to correct - taxes withheld last year.