I already filled out a federal tax return with another online service. How do I only submit a state tax return with Turbo Tax?

You can't. State cannot be done separately from federal using TurboTax. You would have to prepare the federal return exactly as it was prepared when you filed it with the other service in order for the information to flow to the state return. Why are you not using the same service you used to prepare your federal return?

Thanks! I ideally wanted to use the other service but my payments for the state return charges keep encountering issues with them. So it would be alright for me to file my federal a second time, this time through TurboTax, just so I can submit a state return through TT as well?

You can't file a second federal return. You should be able to choose to print and mail your return and then don't mail the federal return.