If I am moving to NC in July, as a non W2 employee, how do I estimate my quarterly state taxes? I've calculated my estimated quarterly taxes for my current residence.

Well, for 2024, you don't have to schedule right away.

1)  your NC quarterly payments are due before 15 Sep for income from when you arrive in July, and 31 Aug.  So you have 2 weeks to guesstimate your net gain, and pay in (About 5% of the net gain).

2)  For 1 Sep to 31 Dec of 2023, you pay another ~5% before 15 Jan of 2025.   

3)  Of course, if you have other income from investments, especially Mutual Fund distributions that usually happen in December, you might add some more to that final NC payment to cover those $$.

That will get you started...then you can readjust your payment amounts the nest year depending on whether you think you overpaid by using the 5% of net.   

(The NC income tax rate is actually 4.5% for 2024 and 4.25% for 2025, but it's almost impossible to pre-figure the exact amounts to pay until you accurately figure in your other income sources too, and your exact business expense deductions that produce your net SE income)