Hello Zach,
From your description, my understanding is that you have a shared mailbox which still receive amounts of emails Inbox and as time going these emails take much storage of it?
If so, based on my experience, there’s not many efficient way to manage these emails, one of which is Inbox rule. You could login Outlook Web App with your own account, then click “My account” ->Open another mailbox -> enter shared mailbox’s address -> then you will enter shared mailbox view, in this view, click Settings-> View all Outlook Settings-> Inbox rule:
You can set a rule to delete emails or manage them with sub folders. Here’s one related article: Use inbox rules in Outlook on the web. Use Forwarding settings to forward emails: Turn on automatic forwarding in Outlook on the web.
You could also use Outlook desktop app to back up emails for shared mailbox by Export feature: Export or backup email, contacts, and calendar to an Outlook .pst file
Besides, by default, a shared mailbox has 50 GB storage, you could assign it a license so its size can be increased to 100GB: Mailbox storage limits
Archive mailbox can also be used on shared mailbox with a license: Archive features in Exchange Online Archiving
If you have different scenario, welcome back and share with us.
Best Regards,
Anna