There was a known issue in D:OS EE where the game could hang on the loading screen trying to start a new game, and give an error about network initialization clicking on Join from the main menu, possibly caused by a too strict firewall, network drivers or other problems with a network card.
In that case, it would help to reset your network adapter. One person has had a similar problem in D:OS 2, which this helped with, though they were not getting the connection errors.
- 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:
If that doesn't help, try exiting out of the Steam client, and starting the game directly from the '..\SteamApps\common\Divinity Original Sin 2\bin\EoCApp.exe' program file, by right clicking and running as administrator.
If you start the game from the executable with the Steam client running, it should be able to connect to make Steamworks functions available (the multiplayer lobby, friend invites).
Try creating a new profile, and see if that will let you start a new game.