We selected direct debit to pay our state taxes but turbo tax keeps emailing us saying we need to mail in our payment. Should we wait and see if the direct debit works?

What did the screen tell you in Step 2 about how you are supposed to pay your state?  Many states want to be paid directly.

**** Make sure you set up the payment correctly ****
Nice that they are sending out email reminders now ... that should reduce the amount of after season posts of upset users that missed those instructions in the program ....

Check the printout or PDF of your return; look for the federal and state cover sheets with the Turbotax logo.   

If you owed tax, they will show the payment information and how you decided to pay.  

>If you indicated pay by mail, then you need to mail in a check.  

>If you selected credit card, it will probably be processed in a couple of days.  

>If you selected debit from your bank account, that information is passed on to the state and IRS and they will do the debit when they process your return information -- usually 1-3 weeks for e-file and 3-4 weeks if mailed in.  

> If you also owe the state ... read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.  https://ttlc.intuit.com/questions/1900884-what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe