SOLVED (for me)
Hopefully this solves your issue, too.
I hired a server admin to dig into this issue. It turns out osTicket does not properly handle email account passwords that contain special characters. Removing all special characters solved the issue.
For example, the password "R$t8#-hM" will fail. Password "We8L3Gb" will work.
Hopefully this will be corrected in the next version.