I have essentially the same issue, although I'm on LInux, and I want to upgrade from free to Studio at the same time.
Something unrelated killed my host OS installation and I took the easy route to fixing it: buy a new SSD, make a fresh OS installation, copy all the files back into place from the old drive.
The directories / files I've noticed so far are as follows
In "user space":
$HOME/Videos/Resolve Project Backups
$HOME/.local/share/DaVinciResolve
$HOME/.local/share/Welcome to DaVinci Resolve
$HOME/Documents/BlackmagicDesign
And the system installation:
/opt/resolve
/var/BlackmagicDesign
What I did was:
Copy all the user-space files into place (along with all my media etc, though those were already in place since they were on other drives that did not get replaced--I simply ensured they were mounted in the same place)
Install the new Resolve Studio
I noticed that the contents of the directory:
~/.local/share/DaVinciResolve/Resolve Project Library
was essentially empty, but originally had a bunch of stuff in it. That stuff had, apparently, been overwritten by the installation process.
I moved the new directory to a different name (so I could recover it if necessary) and then copied the contents of that directory back into place from my old drive.
Restart resolve, and there were my projects, all working, with media and everything
Hope this helps