Do I need covid-19 test to travel NY to North carolina by the flight

No

Not for domestic flights. But as a courtesy and to make sure you and people around you are safe, good to get one within 72hrs of the flight

Much better to take one

No. It really is a simple as that.

Not entirely necessary but strongly recommended. The only mandatory requirement is to fill the NY's travelers form.

I would get one before coming to NY.