Hi NA,
From your description, I understand emails get stuck in Outlook Outbox. I suggest you try creating a new profile https://support.office.com/en-US/Article/Create-an-Outlook-profile-f544c1ba-3352-4b3b-be0b-8d42a540459d?ui=en-US&rs=en-US&ad=US to see whether there is any improvement.
If the issue persists, because this kind of issue is related to the Outlook client while our forum focuses on Office 365 Online services with limited resources on the Outlook client questions, I recommend you post this question in the Outlook forum below for professional suggestions:
http://answers.microsoft.com/en-us/office/forum/outlook.
Your understanding is appreciated.
Regards,
Xinyu