No, TurboTax doesn't automatically file your state return, you must walk through the state interview questions and choose to e-file it.
Please see this TurboTax Help Article for instructions on how to prepare and e-file your state return: How do I e-file my state after I already filed my federal?
Also, you can check your filing status by following these instructions: TurboTax Tools: Where's my Refund?
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