Repoint your old-domain.com MX record to the new IP.
Tell you new Exchange server to be authoritative for the old-domain.com inbound email.
Double-check your new domain to ensure that the default email addresses are the new domain.
Add the old-domain.com to the recipient policy.
The dialogue box is going to ask you if you want to apply this to all recipients.
At this point you may want to say “no” because you do not want all users to have an address of firstname.lastname@example.org.
However, if the number of users is large, you are going to want to figure a way to get the old-domain.com addresses onto the folks to whom you want the old-domain.com email to go to.
You can just let everyone get the old-domain.com smtp entry, and you will be OK, but your policy may dictate otherwise.
Leave the old server up for a time until you KNOW the MX record is repointing and that the new server is delivering old mail as desired.
Then UNINSTALL the old Exchange. Don’t just turn it off…uninstall it.