Exim refuses to send and accept email once the server passes the configured load value.
Increase deliver_queue_load_max and queue_only_load in WHM»Exim Configuration Manager»Advanced Editor to a number greater than the server load average. You can check the server load averages in the top right corner in WHM. Choose a number higher than the highest of the three numbers.
If the emails are still not sending, you may need to clear the retry the Exim retry database as well.
Clear Exim Retry DB to bypass "retry time not reached" after fixing an issue with delivery