Currently, Amazon Music not providing any public API to access to an user Amazon Music account data.
While there's no public API and we didn't found a secure workaround to access to Amazon Music user datas, we are not able to move datas TO Amazon Music.
But we are still able to transfer an Amazon Music playlist to other services by using the "share link" of a playlist.
You can see a thread about this request to Amazon here => Forum Amazon Developer
Help Soundiiz by writing on Twitter to @AmazonMusic and asking to open an API for Amazon Music.
Be careful if you are seeing an app which promises you to be able to move to Amazon Music. These apps could use iFrame method to collect your session cookies / credentials from Amazon which is an illegal and dangerous method (accessing directly to your payment method and can also change delivery address).