How did you "move" the DLs? Are you still in Exchange Hybrid mode and are there send connectors on-prem that route to 365?
You don't need a contact necessarily. If the group email address is:
firstname.lastname@example.org and If you have a send connector on-prem that is set for the address space of
mydomain.mail.onmicrosoft.com , then all you need to do is have the SQL server alerts go to email@example.com and have the SQL server send to that through the on-prem Exchange Server. Then Exchange will route that to 365 and it should work.
Ensure the Exchange Online DL (firstname.lastname@example.org) allows messages from unauthenticated senders as well:
Use this section to manage who can send email to this group.
Only senders inside my organization: Select this option to allow only senders in your organization to send messages to the group. This means that if someone outside of your organization sends an email message to this group, it will be rejected. This is the default setting.
Senders inside and outside of my organization: Select this option to allow anyone to send messages to the group.
You can further limit who can send messages to the group by allowing only specific senders to send messages to this group. Click Add Add icon and then select one or more recipients. If you add senders to this list, they are the only ones who can send mail to the group. Mail sent by anyone not in the list will be rejected.
To remove a person or a group from the list, select them in the list and then click Remove Remove icon.