What is the best way to move my Amazon Music to iTunes so I can listen to it on my Apple Watch

You currently can only download Apple Music to your Watch. You can stream Apple Music if you have a cellular Watch.

https://support.apple.com/en-gb/HT204691

I don't use Amazon Music, but to be able to listen to music on the Apple Watch, it has to be downloaded to the Apple Watch. You can do that through iTunes. Does Amazon support provide any information about downloading music to iTunes?

I can download music from my iTunes library to my Watch as well. However, I would guess that if Amazon Music is a service like Apple Music, I would agree that would be a problem, but I don't use it so I am not familiar with how it works, which is why I made that comment in my response.