Dear Cindy Sarreshtedari,
Good day!
Thank you for posting to Microsoft Community. We are glad to assist.
As per your mentioned description about "Shared Mailbox - Cannot send outgoing email". To understand the situation and be able to offer you relevant suggestions, we would need a little more information from you. Kindly help us in providing the following information.
- May I know your current environment? Is It Exchange Hybrid, on-premises exchange (Exchange server) or pure cloud?
- May I ask whether shared Mailboxes were inserted manually or through auto-mapping?
- May I please confirm that you get same situation in both outlook for Desktop and Outlook on Web (https://outlook.office.com)? To open shared mailbox on outlook on web kindly select Account Manager--> Open another mailbox-->Enter that shared Mailbox-->Open

- May I please confirm that your users have Send As or Send on Behalf permission on this shared Mailbox? Shared mailboxes in Exchange Online | Microsoft Learn

- Would you mind to kindly provide the screenshot of the outlook version are you using? What version of Outlook do I have? - Microsoft Support
- Would you mind to kindly provide the entire screenshot of the error you are experiencing via private message for further analysis and the privacy of your data? For data security reasons, please provide some above requested information in private message.
Here is the steps for private message: Click on Account manager-->my profile-->View private message. (See below screenshot) 

This requested information will help us to narrow down the scenario and find the right troubleshooting.
We are looking forward to your response! Thanks for your precious time. Have a nice day!
Best regards,
Inema | Microsoft Community Moderator