No, you would not send it back through TurboTax.
Yes, you should include a check for whatever amount is owed.
As for how to proceed with the 1040NR and 1040X, you would be best to get that information from Sprintax as you may not need to pay the full amount back. You may not owe your entire refund back, but you will not know until you take care of the amendment with the correct information.
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