Is it possible to file previous year state taxes with Turbotax if I've already filed my federals with them? (Chose to "delete" my state return at the time?)

Like for 2017?   You need to download the .tax2017 file from online and install the 2017 Desktop program to continue.  You will have to print and mail prior year returns.

How to finish a prior year online return   https://ttlc.intuit.com/community/prior-year-return/help/how-do-i-finish-a-prior-year-return-that-i-started-in-turbotax-online/00/26322

Or you can get the blank forms from the state and fill them out by hand.

Yes but it will not be free.   You  will have to purchase and download desktop software for each of the years you need to prepare a return for.  The software can only be dowloaded to a full PC or Mac--not a mobile device.

To finish a prior year return started in TT online:https://ttlc.intuit.com/questions/2559403-how-do-i-finish-a-prior-year-return-that-i-started-in-turb

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS and/or your state received the return.

Another option--Get your printed federal returns in front of you, and then get the paper forms to prepare the state returns by hand and mail them in.

https://ttlc.intuit.com/community/state-taxes/help/how-do-i-contact-my-state-department-of-revenue/00/25573