Are you shutting down all non-essential programs before starting the game, especially anti-virus and any backup programs that may be monitoring the Documents folder? A handful of people have reported that Comodo Cloud Antivirus created a sandbox for the game (in one case installed unnoticed as part of a browser download), so any changes to the My Documents folder were done to a separate folder, and after restarting the game, everything would be back to how it was before the previous session.
Check the save folder to see if there are any saves there, not showing up in-game. Each save is a folder in the '..\Documents\Larian Studios\Divinity Original Sin Enhanced Edition\PlayerProfiles\<ProfileName>\Savegames_patch' folder. You can try searching your hard drive for *.lsv files, if there is a chance a security program created a sandbox for the game (in the case of the program mentioned above, each new session was a different folder under the C:\CCAV folder).
If you start the game, switch to windowed display mode and switch to Windows Explorer, you can check the savegames folder before exiting the game, and optionally copy or zip the saves to another location. This may help in determining what is happening to the save, and you should at least be able to restore the save if it is removed.
Try exiting out of the Steam client, and starting the game directly from the '..\SteamApps\common\Divinity Original Sin Enhanced Edition\Shipping\EoCApp.exe' program file, by right clicking and running as administrator, or similarly for Galaxy and the GOG version install path.
Try doing a clean boot and then test the game. Click Start, or hit WinKey-R, type in msconfig and hit enter; in the General tab, click Selective Startup, uncheck Load startup items (if required) and leave Load system services and Use original boot configuration options checked. Next, click on the Services tab, check the box to Hide all Microsoft services, then click the Disable All button (maybe make a note of which are currently enabled/disabled), then click OK and reboot the computer. Run msconfig again to switch back to the normal boot configuration.
Try creating a new Windows administrator user account, switch to that account and try starting the game from there, directly from the executable to test.