Using Classic Outlook on Windows for personal email, calendar, and contact management
Hello Colin,
What permission did you assign? From the error message, it seems that you only assigned the Full Access permission, so you can add the shared mailbox to your account and view the emails, but you are not able to send the emails as the shared mailbox.
Please sign in Office 365 and go to Exchange Admin Center(EAC)->recipients->shared, click and edit the shared mailbox. Then go to mailbox delegation, add your account to the Sent As permission list. Then wait a few hours to take effect. Then you will be able to send emails as the shared mailbox. In Outlook 2016 client, click From and choose Other Email Address and type the address of the shared mailbox.
Regards,
Rick