If my federal and state taxes were accepted, then do I still need to send in a payment voucher?

How did you select to pay it?  Did you enter your bank info for the IRS to pull it out?

You can pay the IRS directly at

https://www.irs.gov/payments

For a state tax due

Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo.  If you owed tax, it will show the payment information and how/when you decided to pay.  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/community/refunds/help/what-are-my-options-for-getting-my-state-refund-or-for-paying-any-state-taxes-i-owe/00/25954

Be Careful ... if you owe and choose to mail the payment there will be ONE 1040-V for the 2020 balance due to mail in with a payment.

HOWEVER since you owed on the 2020 return  the program may have populated FOUR  1040-ES forms which are for the 2021 quarterly estimated payments if you need to make them.  These will NOT pay the 2020 balance due they are ONLY for pre paying your 2021 taxes in advance so you don't owe again next year.  These are not mandatory simply suggestions.  

Q. How to change payment method after my e-filed taxes were accepted?

A. You can't change it.

You selected a method of payment before pushing the file now button.  That's the method you should use.  It sounds like you selected mail in the payment with a printed voucher.  Read the "filing instructions" TurboTax prepared to verify that.