Help & troubleshooting for channels on your Roku device, including adding/removing channels, logging in to, authenticating, or activating a channel, channel-specific playback issues, assistance contacting channel publishers to report issues, and adjusting channel-specific settings.
Did you subscribe directly to AMC+? Or did you subscribe through The Roku Channel?
If you subscribed through The Roku Channel, it shouldn't be asking you for your email/password from within the Roku Channel. That would be something the standalone AMC+ app would do, and a subscription through The Roku Channel won't work on the standalone app; you have to watch from within The Roku Channel app.
DBDukes Roku Community Streaming Expert
Help others find this answer and click "Accept as Solution." Note: I am not a Roku employee.