After completing several updates on Exchange 2013 due to another issue, we found that e-mail messages were going to the Drafts folder instead of the Sent Items folder. In addition the server was not receiving any messages either.
After further investigation we found that the Mail Transport service was failing, not immediately, which was why we didn't notice it after it was started. In this particular case the client had a secondary SMTP port configured for relay messages but that port was setup on both the Exchange 2013 CAS and Database servers but only one of the CAS server actually had the IP address assigned on the NIC. Once the relay SMTP port configuration was removed from the Database server the Mail Transport service ran fine and mail flowed through the system correctly.
If you have a similar issue with mail being stuck in the Drafts folder on the client, check the Mail Transport service to see if it is running and check for errors on the in the Event Log to help determine why it might be failing.
We hope that this information is helpful. Please let us know how this has helped you or if you have additional questions. As always Firestone Technical Resources, Inc. is here to help with your computer support issues - "Providing personal service for your impersonal technology."