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Exchange OWA 2013 Emails Stuck in Drafts folders

Like them all, I too faced this issue.

Checked the DNS, group policy, restarted Transport services, checked if any service was in maintenance mode - all failed.

I opened the services mmc and checked the relevant Exchange services. In my case the 'Microsoft Exchange Mailbox Transport Submission' service was not running. 

I started the service and emails started flowing again.

Hope this helps!