In full screen mode only..
Disabling enhanced precision (acceleration) in Logitech Gaming Software does not work, nor is changing polling rate.
(I 99% assume you've already rebooted your computer and the restarted the game, that often takes care of such strange issues)
This sounds like a problem of the AGS engine rather than a specific problem with this particular game, maybe Chris Jones or the folks at the AGS forums know a fix: http://www.adventuregamestudio.co.uk/forums/index.php
PS: Problems with Logitech mice and AGS seem to be not quite uncommon. If nothing else helps, you can try to uninstall the Logitech drivers
(backing them up of course) just until you're done with this game. An inconvenience in case you're playing other games at the same time that require advanced driver settings but if Technobabylon is the only game you're going to play for the next 2 days or so (shouldn't take longer to complete) then it could be a simple fix for the problem.
Well, I will probably rather play the game in a window where the bug does not maniferst itself, but it diminishes the immersion (having the residual desktop clutter around the game window).