Moving to new state in September

Congratulations on the move @ DebbieF2021.  Moving from a state with no income tax, Texas, to one with income tax, North Carolina, can create challenges, so thank you for the question. 

You would be a part-year resident of North Carolina from some time in September to the end of the year, and therefore, would need to file a North Carolina Part-Year tax return.  See (https://www.ncdor.gov/taxes-forms/individual-income-tax/nonresidents-and-part-year-residents)

North Carolina requires estimated income tax payments if the tax shown due on your return, reduced by your North Carolina tax withheld and allowable tax credits, is $1,000 or more regardless of the amount of income you have that is not subject to withholding.

Since for Tax Year 2023, the North Carolina individual income tax rate is 4.75% (0.0475) if your Self-Employed income is not significant, it is likely that you would exceed the $1,000 tax that requires the payment of estimated taxes.   If your Self-Employed income is significant during your part-year residency you may consider making estimated payments.