You can request your federal transcripts by mailing in form 4506-T. (The online system only goes back 3 years.)
https://www.irs.gov/forms-pubs/about-form-4506-t
But that will only give you your income and federal tax information, it won't tell whether or not you filed a state tax return.
In the end, only you are ultimately responsible for preserving your own tax records, including proof that you filed, copies of your returns, your refund, etc.
Even though Turbotax only keeps the last 3 years (if you used Turbotax online) that might help you prove you filed Illinois taxes for the last 3 years when Illinois says you haven't filed since 2008. Maybe they have some kind of a record keeping problem with your account.
The entire situation seems odd, you may want to work with a professional. If they are billing you now for 2015, what about 2016, 2017, 2018, etc? If they don't have you on file for those years, are they going to bill you later? It might be worth it to pay someone to fix the entire situation for you all at once.
