Realized we can't send out from an alias email address using Office 365 / Online Exchange. Well, we can choose to do so but external recipients see our primary SMTP address. I made the alias primary SMTP and waited a bit but same thing. I then removed the alias and created a Shared Mailbox with the alias as the primary SMTP and waited a bit but surprisingly external recipients still saw the primary SMTP from our "main" account.
For example, I removed smtp:firstname.lastname@example.org from my user / email account and created a shared mailbox with email@example.com as the primary SMTP email address. I received a UPN error when trying this so I simply changed the display name on the shared mailbox to "Me Alias Domain" from just "Me" (which matched my primary account). That worked to create the shared mailbox and the me@aliasdomain email address could be typed into Outlook and used to send an email. Internal recipients saw the alias domain but external recipients saw the primary domain email address.
Any ideas how to send out from Office 365 using an alias without a third-party tool are much appreciated.