When you first login, note that the first prompt is asking for the current password, not the new one. It is the one that is emailed to you. If that is entered incorrectly, you will be logged out and the password will not have changed. The flow looks 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: 

If you’re still having issues, you can reset the password in the DigitalOcean control panel:

Of course, if you use SSH keys you won’t need to worry about the password at all. Check out:

How To Use SSH Keys with PuTTY on DigitalOcean Droplets (Windows users)

I have too faced this problem once.. Fortunately I was in initial trial and error stage.. not live, all I did is wipe it and rebuild the image :)

Hi, I just created a video to show how to do one click connection with root and sample user with putty on Windows :

Here is the video (20min) :
https://youtu.be/Iq3rr5yCAmk

Have a great success with your droplet !