Check Whether You Have Access to a Shared Mailbox
First, please make sure that you have access to the shared mailbox. If you’re not familiar with how to check your access permission, please follow the instructions in the link below:
Email: Accessing and Managing Secondary Mailboxes
If you do not have access to a shared mailbox, please submit a ticket to request access using the link below:
Account - Update Access and Permissions
Add Shared Mailbox to Newsletter as Owner
On the "About" page of your newsletter, click on "Edit" button:

Then, type down the name or email address of the shared mailbox to the "Owner" section:

After click on "Save", the shared mailbox is added to the newsletter as co-owner:

Send Newsletter on Behalf of Department
Now, open the newsletter you want to send, click on "Next" to navigate to the "Send" configuration page:

Click on the drop-down arrow on the "From" box and click on "Add non-author sender":

Enter the shared mailbox to the pop-up window and click on "Add" button:

The newsletter is now sent on behalf of the department, not individual:

Now, the "Send" button on the op right now change to "Send As" button, click on "Send As" button to send:

The newsletter is sent from department as a result:

Related articles:
How to use Outlook new Newsletter feature
Email Distribution Groups
Finding Email Distribution Groups (DGs) and List of Auto DGs