I deactivated my Facebook account. Every time I go to Messenger to continue playing a game, it wants me to log back into Facebook. Then I receive emails thanking me for coming back to Facebook. How can I keep Facebook deactivated and still continue games on Messenger?
No, that is not possible as of now. Because games are associated with your Facebook account. You can use Messenger without having an active Facebook account just to have conversation with your contacts but you cannot use any other service of Facebook by just login to Messenger.
You will not be able to access Gameroom if you deactivate your account.