I had the same problem, you have to go into your amazon account settings and deauthorize the devices there, as it has a 25 device limit and each rom install counts as one so they fill up quickly.
This is what I had to do as well. Apparently each new Rom I've used on any device I've owned counted as another authorized device. Drove me insane trying to log into the amazon appstore app. Logged into the amazon website and managed my authorized devices. No issues logging on since.
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