Did you get a tracking number from the post office when you mailed it? Did you also mail federal? Federal and state go in separate envelopes to different places.
And Make sure you are using the refund amount from your actual state tax return and not the combined amount Turbo Tax showed you at the end. That was for federal and state combined.
The tax due check has not cleared your bank account? Guess you better cancel that check at the bank and mail it again. Maybe include a copy of the post office receipt so they know you sent and paid it on time. It might help. Or if they send you a bill for Penalty and interest for paying late send your post office copy then.
Or the best fastest way to pay would be to pay directly on the state's website.
For completeness, if you haven't already done so, you may wish to phone the state tax agency and ask them if they received your return before submitting another one. But normally I would think they would cash a check fairly soon after getting it.
You haven't mentioned any timeframe that I could find above. How long has it been since you mailed the return? Was it back in April or more recently?