To avoid getting double alerts from other email Apps, you can keep your email account settings as is, and use the old App whenever you like. You only need to adjust the notification and badges settings of the other service.
For iPhone: Go to ‘Settings’ and tap on the ‘Notification’ section. Select the old email service and change the alert style to ‘None’. Then put the ‘Badge App Icon’ and ‘Show in Notification Center’ to off.