Hi,
We have migrated 14 people to Office 365 E1 plan. They have one problem where some emails are received in their Inbox both in Outlook and OWA as drafts. That is, when they open the email to read it there is a From box allowing them to send as the person who they received the message from (this can be an external or internal user). All the names are underlined and if they send it, it goes to their Sent items.
It's not happening to everyone, the only correlation appears to be the affected users are also Blackberry users. They did not have this issue before moving to 365. They are using the BES service provided by 365, and had their previous email accounts removed before being added to BES. They have quite old devices (Bold 9700 from what I remember).