Hello Mike10853,
Thank you for reaching out in Apple Support Communities. Even with Do Not Disturb turned off, do you see the icon of a crescent moon for Do Not Disturb in the status bar of your iPhone? This is what it looks like:

Do Not Disturb is on. This setting silences calls, alerts, and notifications until you turn it off. Alarms will still sound.
This is illustration in: Status icons and symbols on your iPhone - Apple Support
If the icon of the crescent moon does not appear in the status bar of your iPhone, let's review the settings for your notifications. Please go to Settings > Notifications > Messages. You'll want to make sure that "Allow Notifications" is turned on and that your alerts are enabled with "Badges" toggled on. This will also apply for the notifications of the Mail app in Settings > Notifications > Mail.
Once that's confirmed, please restart your iPhone and see if you're able to receive notifications of your apps.
Kind regards.