It shouldn't be asking for your credentials again after you've changed the password. The flow should look like:
You are required to change your password immediately (root enforced)
Changing password for root.
(current) UNIX password:
Enter new UNIX password:
Retype new UNIX password:
Note that the first prompt is for the current root password. The one that is emailed to you. If that is entered incorrectly, you will be logged out and the password will not have changed.
If you're still having issues, you can reset the password in the DigitalOcean control panel: