We are doing a Migration from on-premises Exchange 2010 to Office 365 and we have installed Azure AD connect already.
I am slowly doing the Mailbox Migration, Question stuck in my mind is what do I do with Shared, DL and Meeting rooms? How do I migrate them?
Do you have a hybrid environment? If so, you can use AADConnect to synchronize the on-premises DLs with AAD so that they appear in the cloud GAL. The groups will have to be managed on-premises because they are "owned" by that environment.
Office 365: How to migrate on-premises Resource (room) mailboxes https://social.technet.microsoft.com/wiki/contents/articles/31163.office-365-how-to-migrate-on-premises-resource-room-mailboxes.aspx
shared mailbox: The procedure is similar to how you migrate a regular user mailbox, this needs to be done in Exchange online admin center:
In Exchange online Admin Center > recipients > migration > + > Migrate to Exchange Online > choose Remote move migration > Next > under "Select the users you want to migrate" choose your on-prem shared mailbox.
As for your Distribution Groups, those will sync to Azure AD/Office 365 with the rest of your users and groups. If Azure AD Connect is installed and syncing to Azure AD/Office 365 then you should see your Distribution Groups in Office 365 under Groups.
As for your shared mailboxes and meeting room mailboxes, you'll migrate those just like you migrate your user mailboxes.