If you e-filed your return, you do not need to mail a copy.  If you did not e-file (you are filing by mail), you have to mail your return before they will send you your refund.

I received an email from turbo tax saying that the last step was to mail in a copy of my return to the state. 

I e-filed.

Check our E-File Status Lookup Tool to get your current status.

Click the links below for more options

Federal Return Refund Tracking (sent to the IRS)

State Return Refund Tracking

If you have not yet submitted the state, you will need to mail in the state. It is possible to e-file the federal and not file the state.