Do I also need to send paper copies of the tax returns to the IRS and the state?

Not if you e-filed your return.

Not if you e-file.  If you e-file, you would only need to send documents that TurboTax tells you to send IF they tell you to send anything.  For the most part, when you e-file, that is all you have to do.  E-file and you are good to go once you get the message saying your returns have been accepted. 

If you e-filed, it was all done electronically and you do not mail anything.   Keep your documents and a copy of your return for your own records.