Hello @Ralph Taylor Welcome to Microsoft Q&A community forum!
I understand that you must have followed the correct process to move the data from Server 2016 to Server 2019. However, the old server registration cannot be used to continue backups of the new server. Server registration to the vault is a mandatory requirement to perform backups of on-premise machines to Azure. That said, you need to install "Microsoft Azure Recovery Services Agent" (MARS) on the new server and register it to the Recovery Services Vault (RSV) just like a new machine.
Reference doc link - https://learn.microsoft.com/en-us/azure/backup/backup-azure-file-folder-backup-faq#what-happens-if-i-rename-or-change-the-fqdn-of-a-windows-machine-configured-for-backup-
----------------------------------------------------------------------------------------------------------------------
If the response helped, do "Accept Answer" and up-vote it