If you are asking which version of TurboTax to use for your 2020 tax return you are going to need either online Self-Employed or any version of the CD/download. Those are the versions that can prepare the schedule C you need for your self employment income.
You do not file quarterly tax returns----if you are making estimated quarterly payments you can use the 1040ES vouchers that will print out with the printed 2020 tax return, or you can just pay directly on the IRS site to make your quarterly payments.
It sounds like you have two state returns to prepare for NY and NC----sounds like part year for each state.
https://ttlc.intuit.com/questions/1901110-do-i-need-to-make-estimated-tax-payments-to-the-irs
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