by Ken Task.
Ok, tried lots of things ... but please answer this question:
Does your site have links like:
http://.../course/view.php?id=377&MoodleSession=b3i20qk970m1t92b2t2v1v2t90
on it's front page? Yes or No?
Something either in config of apache server (rewrite rule?) or Moodle OR network (as Dan has also suggested) is injecting (tacking onto the end of them) the '&MoodleSession=blah blah blah' in the URL's.
Mind providing the URL to your Moodle front page?
'spirit of sharing', Ken