Dear Bulbous Accounting,
Good day! Thanks for posting to Microsoft Community. I am glad to assist you.
Based on your description, may l know if you are using Microsoft 365 for business subscription?
If so, you can enable online archiving service which includes 50 GB additional archive space. Exchange Online Archiving service description - Service Descriptions | Microsoft Learn.
Note: Only admin can enable Online Archiving service ; if you are not, please kindly contact your admin.
After enabling it, you will see a new folder named Online-Archive in your Outlook Manage email storage with online archive mailboxes - Microsoft Support. Move the old emails to that folder manually or create a retention policy for it which can move old email to online archive folder automatically.
Create a Retention Policy in Exchange Online | Microsoft Learn

If you have already used Online-archive mailbox up, you can choose to export old email from your primary mailbox. Export or backup email, contacts, and calendar to an Outlook .pst file - Microsoft Support

For any other questions about above method, please do not hesitate to contact us. Thank you so much.
Sincerely,
Tammy | Microsoft Community Moderator.