Hi Scott,
The screenshot of security groups you provided above is a group of SharePoint Online users that is granted access to specific SharePoint Online resources by an administrator. For example, an administrator can create a security group to grant a certain group of people access to a SharePoint site. Therefore, after you created a distribution group and selected "Make this group a security group", it will not displays in security groups page.From the description, I see that you've converted one mailbox to shared mailbox, but there an issue occurred when you give permissions to the distribution group. Please close Windows Powershell and right click and select "run as administrator". After that, try to connect with Exchange online to check if it works.
Please note that, you need to assign the security group the FullAccess permission to access the shared mailbox and assign the security group the SendAs permission to the shared mailbox
For details, please refer to the link below:
http://help.outlook.com/en-us/140/ee441202.aspx
In addition, if the shared mailbox is larger than 5 GB or you need to enable archive mailbox for the shared mailbox, license is required for the shared mailbox.
Thanks,
Aaron Yuan