I am not receiving all the mail sent to my account. I have 3rd party or departmental forwarding set up in the Electronic Directory Editor (EDE).
Mail sent by another Exchange user will be delivered to an Exchange account if it exists and will ignore any EDE forwarding rules.
Example: Exchange user A composes an email in Exchange and uses the "check names" feature to send an email to User B. User B has an Exchange account and forwards all their mail to Gmail via EDE. User A's email will be delivered to User B's Exchange account. In order for all mail to be delivered to the account in EDE, you will need to set up an inbox rule from the Outlook Web App. See Office 365, Email, Exchange, Forwarding Exchange Email to Another Account.