My old machine died. How do I go about moving my Directory Opus license to my new machine?
You don't need to do anything special; just install Opus on the new machine as you would normally.
If you can still access the drive on the old machine, you can transfer your configuration if you wish to. See Locating the config files by hand in the Config backup FAQ.
But what about the license? I installed it on my new machine, but it keeps prompting me for my license. I have the email, but it says it's already been used. Can I somehow make my license valid again without being able to run my old copy?
If the licence from your email says it has already been used then you've probably found the wrong email and used an earlier free evaluation licence, rather than the full licence which you later bought.
If you've found the right licence but Opus is still prompting you about it then the licence may not have been properly installed. Please see this guide for troubleshooting.