Don’t use the default overdue notification. Create an additional notification with the overdue event specifically for this action. You will also need a custom template that indicates that the request is closed. You can always put any content you want in custom notifications, no need to indicate that the action was overdue.
It makes sense in this case to send the notification with the “Overdue” event because the request will always close in that case.
Regarding your second concern, the Workflow’s engine service checks for new overdue actions every 3 minutes by default so it doesn’t overuse the server. You might need to wait a bit before seeing the overdue update.
If that’s not the case, check the engine service to see if it’s running after the update.