There are several ways to address this. While the mailbox size can be increased, there is a technical limit (Microsoft Outlook limit) so here are a few additional ideas:
Archive email
Archive older items automatically
Archive or mute Gmail messages
Also, Print Email or Website Information to PDF. This allows you to retain the information without clogging up your inbox.
When you print to pdf, saving them to Google Drive is a great option. The easiest way to save them directly to Google Drive is using Google Drive For Desktop.
You can save files to the X-Drive, but storage space is limited and only accessible on campus or using VPN. Google is FERPA compliant, for decisions where to store. File Storage and Sharing Options: Google Drive vs. X-Drive vs. OneDrive
TWU offers several instant messaging tools which can be used as alternatives to email. See Instant Messaging (Google Chat and RingCentral)