Are you shutting down all non-essential programs (especially anti-virus) before starting the game? Firewalls have also caused conflicts, including this issue after the last update, with some wanting to re-verify permissions for the game.
Try switching to windowed mode, or Alt-Tab out of the game to see if there are any popup conformation windows from your anti-virus or firewall program.
Problems with network drivers or a network card can also cause these symptoms, and when clicking on Join from the main menu, an error about network initialization. In this case, it should help to reset your network adapter (instructions below, if required), but since the game was working fine, an issue with a firewall program seems more likely.
- click Start (or Search in Win 10), type cmd into the search box, right click Command Prompt and select run as administrator. - type the following command and hit Enter netsh winsock reset - when that is done, there should be a message 'Winsock reset completed successfully' - type exit and hit Enter, then reboot your computer. For more details, see: https://iihelp.iinet.net.au/Resetting_Network_Adapter_(Winsock_Reset)
To verify local files: in the Steam library, right click on Divinity Original Sin Enhanced Edition and select Properties, switch to the Local Files tab and then click on the 'Verify Integrity of Game Cache...' button. With the GOG version, in the (optional) Galaxy client, select Divinity: Original Sin Enhanced Edition in the left column, then click on the More button, and in the Manage menu select 'Verify / Repair'.