A few issues with this situation. I wouldn't move the seasonal stuff to the home page if they have their own pages, and the way it is being done here is creating duplicate content. I have never seen this done before where someone if moving an internal page to the home page and looking to have a canonical from the home page to an internal page. I would probably recode it so that it is an item featured on the home page and not the full home page.Adding in a canonical and changing out the canonical will most likely do more harm than good.
A better situation might be to redirect the home page to that seasonal page, but I would still venture towards having it as a featured item on a home page that is in fact the home page.
As for the internal stuff, you have duplicate content. Just changing a few words on the page doesn't reduce it that much, and your descriptions aren't really changing at all. Since you have very few colleges I would create individual pages for each college, make the content different by including the teams, campuses, etc.
Usually anything after a ? in the URL is considered the same URL. Granted on Page 6 (for me) using the site command:
site: dormroommovers.com
Shows the full listing of schools.
Using a canonical to the page:
/How-Ship-To-School-Works.php?School=Goucher College
I would advised against, and am not really sure it would do what you want.
If you want to still use the pages with the duplicate content you can use an htaccess and rewrite the individual colleges to that script, so you don't have to recode anything. you could then canonical to those versions. So using the above example change it to:
/move-to/Gougher-College/ or /Gougher-College/ etc.
RewriteRule ^/Gougher-College/$ /How-Ship-To-School-Works.php?School=Goucher-College