My question is how to maintain local Active Directory for computer
accounts and logons and keep passwords in sync with Office 365?
If you're already using Azure AD Connect then you don't need to do anything, you're already doing it. Keep using Azure AD Connect like you are now. If you're using password hash sync then the password hash for each user account will sync to the Azure AD/Office 365 user account. If you're using passthrough authentication than your Office 365 users will authenticate directly with your AD.
A couple of points, based on your comment:
You probably want to install Azure AD Connect on the new server in staging mode. That way it's ready to take over once you decommission the old server. You can find more information about staging mode in Microsoft's online documentation.
The Hybrid option is used/needed for when you are going to use the Exchange Hybrid configuration to migrate mailboxes from an on premises Exchange Server to Office 365 and you're going to have mailboxes on both sides for some period of time. After your on premises mailboxes have been migrated to Office 365 you will need to maintain a small footprint Exchange server on premises for managing Exchange related attributes for your users, groups, and contacts.
Azure Active Directory (AAD) comes in a number of editions. Azure AD Basic comes with Office 365 and is the "free" edition of Azure AD. So... you already have Azure AD. You don't need to pay anything additional for it. If you want more advanced Azure AD features you can purchase one of the other Azure AD editions.
Azure AD DS (Azure Active Directory Domain Services) is another animal altogether. For now, you should ignore anything related to Azure AD DS.