This will allow you to sign into Amazon apps with different accounts.

You need to be rooted.

With RootExplorer navigate to com.amazon.dcp.apk in /system/app

Rename com.amazon.dcp.apk to com.amazon.dcp.apk.bak

Reboot

Now you'll be able to sign into you account within the amazon apps instead of it automatically using the account your fire is registered to.

This worked great for me but I assume no responsibility if you mess something up.