I use Amazon Unlimited and access the music from the Sonos app. I use the search tab which works well. Just opened the actual Amazon music app and all my speakers show up so don’t know why yours don’t - sorry. 

It’s been a while since I looked at the settings, but my understanding was that the link between Amazon music casting and Sonos was through the  Sonos Alexa skill.  Are you using Alexa?  If so, do you see the Sonos speakers in the Alexa app?  Have you tried removing and readding the Sonos Alexa skill?

I do not use Alexa and have no interest in using Alexa.  Do you have to use Alexa?  Seems like a pretty poor set up by Amazon if that is the case.  I guess I will have to stick with Spotify, which is fine, just wanted to see other options.  

I don’t know for sure whether Alexa is required, but if it is required, you can create Alexa account and link it to Sonos….but not enable Alexa voice control on any Sonos speaker.

https://support.sonos.com/s/article/3248

How do I control Sonos from the Amazon Music app?

You can control Sonos directly from the Amazon Music app by following the steps in Amazon’s article about casting. Note that the Sonos skill must be enabled in your Alexa app, even if you are not using Alexa with Sonos.