I have forwarded this on to the IT people, but here is the bits I could do and the results.
First of all, I’m not sure the exact site move procedure, but the old and new sites have the same url. Relative paths should all be the same within the site directory. At some point, to fix another problem entirely, they did a site copy and attached it to IIS on the server (whatever that means!), but the calendar navigation issue started with the original server migration, and was not changed by this.
I did move the contents of /all-in-one-event-calendar/cache, and it was able to regenerate the files it needed and keep working. Within the emptied folder ‘cache’ it created a folder called ‘twig’ and a file called ‘b29149c4_ai1ec_parsed_css’ on my first load of the calendar, so it seems to be able to write to it and create directories. This did not, however, fix the problem at all.
Thanks so much for your help!! If you have any other insights I am all ears, and I will get IT to check out the database.