So after exiting out of the game, div.exe does not show up in the Task Manager? Did you check the Processes tab, in addition to Applications?
Does it make a difference if you exit out of the game by using Alt-F4?
Try right clicking the '..\SteamApps\common\beyond_divinity\div.exe' executable, select Properties, switch to the Compatibility tab and check off the 'Override high DPI scaling behavior' option (set it to Application), maybe run as administrator and disable fullscreen optimization, set a Win 9x or XP compatibility mode and see if that makes a difference.
Try doing a clean boot, start the game from the executable and see if this behaviour persists.
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.
