When a Ticket is Created (unassigned) → Email is send to these 2 share mailboxes…(the 5 Helpdessk admins don’t receive emails).
But are the 5 help desk admins part of the shared mailbox? I guess I don’t understand why the 5 admins don’t receive emails from the help desk directly?

The 5 Helpdesk admins could log into the Helpdesk and Take ownership or disptach
This would still occur with what I’m proposing.
Then the assignee and the creator of the ticket receive email
This is how it’s supposed to work by default.
Enable notifications for the 5 help desk admins, get rid of the shared mailbox accounts, and let the application take care of the notifications.
All this said - the Spiceworks Desktop App is reaching end-of-life Dec 31. Have you considered using the Cloud Help Desk?
More here: What’s going on with Spiceworks Tools & Apps in 2021?