Hi pcaudill,
I want to confirm the following information:
1. What the “Outlook 365” means, is it the Outlook 2013 desktop application or the Outlook Web App (OWA)?
2. Where did you download the “Outlook 365”, was it from the Office 365 Home Premium?
Based on this situation, I suggest you check if the deleted emails still exist in OWA. If yes, this is probably a client issue. Please try to find a solution in Office for Mac Community Forum.
Thanks,
Allen Zhang