Using older versions of Outlook for Mac for personal email and calendar
Sorry to hear you're having trouble.
As well as the information already provided I also had a suggestion. From your original post it sounds like you may have deleted the older emails (correct me if I'm wrong). If this is correct then it may be that for some reason the mails are possibly being saved on the website. I would suggest turning off synching on your app then log in on the website and delete any emails that are no longer required. Then do the same on your app. Restart the Mac before turning sync back on and see if the issue persists.