E-mail is one of the most abused and hackable software tools around. It may be easy to set it up but it takes considerable effort to secure an e-mail system and yours looks like it isn't, infact it's woeful. This is not your fault, it's your organisation's fault entirely for being ignorant and not engaging professional experts to secure it properly.
In your setup an unauthorised e-mail in your Sent folder could be achieved without even using your computer. For example, if your e-mails are stored on a central e-mail server, one can simply log into your POP/IMAP server from another computer using your credentials, which are easily discovered, and then send the unauthorised e-mail. Next time you log into your computer and run Outlook it will look like you sent the unauthorised e-mail.
The shoddy e-mail system you have described and the way it is being shared with other staff means it has too many ends open for abuse. Forensic investigation to cover all possible avenues of abuse would be prohibitively expensive, even for the company.
Seems to me you're dealing with @rseholes who just want to do you. Be sure to let them know that their e-mail system is not secured to worlds best pactices ( IT w@nk words) and can be subject to unauthorised abuse.
Funny how values change over time, 15 years ago I introduced this novelty called 'internet' to an organisation, about 4 months later the proxy logs showed that the finance manager was accessing deviant porn sites. I mentioned it to my manager, she had a direct chat with the finance manager, problem never occured again, everybody was happy.