What? Each tax year must be filed separately. You cannot "save" your state returns to file later with another year. If you used TurboTax to prepare your federal returns for the past years in question then you need to go back to those returns and add the state returns using the software for those years to prepare the state returns. Sadly, since the online software is now retired for any year except 2018, you will need to purchase and download the desktop software for those years in order to use TurboTax to prepare the past year returns. That software must be downloaded to a full PC or Mac--not to a mobile device. And you cannot e-file a prior year return. It has to be printed, signed and mailed.
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.
Your other option, which would be lower cost, would be to get the paper forms from your state's tax website and prepare the past year state returns by hand and mail them in. You will need copies of the relevant federal returns in front of you in order to do that.
Note: You say you came up with this idea "a few years ago"-- it it too late to get a refund for any tax year earlier than 2016.
If you hadn't figured it out, your state revenue department should have made the refund credit for you.
If so, an expected refund would be larger, or a tax due paid would be reduced and refunded to you.
You need to peruse your paper copies of the relevant returns and the final results to confirm this.