No, you do not need to amend your 2020 NC return for wages earned and paid before 2020, and before you became a NC resident.
You should not have received a 2020 Form W-2 for work performed in 2019. The correct course of action would be to contact the employer and request a corrected Form W-2 (one for 2019). Then you can amend your 2019 federal return to report the 2019 income. NC is not involved in this.