The first $45,000 of the appraised value of your permanent home is excluded from North Carolina property tax. The exclusion is a flat amount, so it does not depend on your income or on what the rest of the home is worth. An unremarried surviving spouse can claim it too.
Apply through your local county tax office, not the state. First complete Form NCDVA-9 and have it certified by a North Carolina veterans service officer, then submit the certified NCDVA-9 together with Form AV-9, Application for Property Tax Relief, to the county tax office where the home is.
A certified Form NCDVA-9, a completed Form AV-9, your discharge paperwork, your VA rating letter or specially adapted housing documentation, and proof that you own the home and live in it. A surviving spouse also needs the marriage and death certificates.