That's expected. Exchange does that by design.
Since you have the new Exchange Server up, - IF you haven't done this already - run the Hybrid Configuration Wizard and make the new server the hybrid endpoint with 365. Ensure the firewall ports are up to date to allow this.
Verify things work - SMTP routing etc...
Once done, move all the arbitration mailboxes to the database on new server:
Get-Mailbox -Arbitration | New-MoveRequest -TargetDatabase <NewDB>
Then remove the old server from any send connectors
Then remove Exchange from Add/Remove Programs.
No need to move the monitoring mailboxes. Once you remove the old server, verify things work again.