I have recently migrated my on-prem exchange server 2010 to Office365. We just powered off our exchange servers to test emails and looks to be working fine except when trying to send emails it goes into the sent item folder but the recipient does not receive the email and vice versa.
Also had a look on https://intodns.com/ and shows a warning 'TLD Parent Check'
I have check the Mail Flow and get the 4.4.316, MSG=Socket error code 10061.
I have tested sending email internally and works fine, only sending to external domains is an issue.