The records "burning" thing is vague. I've found that it's sometimes true, but not always. Sometimes it's a "flood".. sometimes it really IS records were lost / destroyed - record keeping definitely was not what it is today.
I'm on a similar quest. My paternal grandmother was born at St. Vincent's orphanage in 1917.
2 years ago when the adoption record laws were changed, I applied to the state to have them search for an original birth certificate.
None was found.
I wrote to Catholic Charities post-adoption services and they claimed that they did not keep much in the way of records "back then" and could not help me other than to tell me when my grandmother was born, baptized, and placed with her adoptive family.
The thing is, my grandmother wrote to Catholic Charities back in 1978 and was given the same non-answer, but they told her that her mother's name was "Margaret". I only learned about this letter after the death of my aunt a few years back (my grandmother passed in 2001).
Back in January I got in touch with the Illinois Confidential Intermediary Service and they said that we can try to use the courts to have a CI appointed to share the actual contents of the adoption record.
The woman at the CI service stated that there might not be much more in that record than I already know (no birth certificate was filed with the state, and St. Vincent's did not record the full name of the birth mother) but that it was worth a shot to see if there was anything useful in the court documents.
So that's the process to try. Submit the paperwork with the IL Adoption Registry, wait to see if they find an original birth certificate, if they do - they'll send you a copy. If they don't, they'll send a letter.
If you get a letter instead of the birth certificate, then you can always try using the Confidential Intermediary service to see if they can pull the court case.
It's a bit of a lengthy process, and does not always guarantee a successful outcome (since birth certificates were not legally required until right around 1917) but it is worth it to try and doesn't cost much, if any, money.
Once you've done the request for the original birth cert., here's the link the IL Confidential Intermediary service. http://www.ci-illinois.org/