You can use any of those triggering events as the date of relocation. You are going to need to file part-year resident state returns (full instructions are in that link) for both Ohio and North Carolina. Next year, you will file one full-year North Carolina resident state return as long as you don't move again during 2022.
When you complete part-year returns, you will need to allocate your income (more info on how to allocate is in that link) between states and the date of residency will come into play for these allocations.
If your employer started withholding taxes too early, filing the part-year returns and allocating the income by state will correct the issue.
You might think that part-year taxes are proportional to the amount of time spent in the state, but that's often not the case. Most states divide the income earned in that state by the total annual income to come up with a percentage. That percentage is then applied to the state tax on the entire year's income—no matter where it was earned—to prorate the tax liability. TurboTax follows the rules for each state.
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