I have a new VM instance in Azure, and by default it comes with C: and D:. I understand that D: is volatile and can be lost at some points.
My application uses drive D: as a working drive though, and stores all my the utilities and process output files there.
I've attached a new drive, but it's F:. Is there any way to change the drive letters so that my attached drive is D: and the temporary one is anything else?
Things what you can do is change the drive letters. You can do it under Server Manager -> Storage Manager -> Pick The Drive -> Change Drive Letter and Path. There can you can change the drive letters.
The litmus test what can perform is rebooting the server and checking for the consistency of the drive letters and data in the additional data disk. Ephemeral disk's data will be gone.
However with the above things in place, there is nothing like modifying your application to use any drive (unlike D: in you case). With that you need to worry about the moving parts and their uncertainty.