Given I work in IT, in a profession that tries to deter fraud, let me give you what I think happened in your case. It appears that your email address was spoofed, for the purpose of sending malicious emails. Email spoofing is actually not hard to do given you have the technical knowledge of how emails are sent and interpreted at the receiving email server.
Using the old method of validation, SPF,(Sender permitted form), your email server as the receiver compares the IP address of its domain host (ex: @XXX)with the origination IP of the email. Only if these match would the email be allowed through, and blocked else wise as spam / malicious email.
This old method is flawed though as it relies heavily on user administration and is not standardarized in how failures are handled. Sometimes, one may not want to block legitimate email, which eases network administration but phishing / spam / become a problem.
A more modern method that is being used in conjunction with SPF is DMARC. DMARC policy is embedded within the DNS information in the email server and tells the email how a message failing SPF validation (Sending origination IP <> receiving domain host IP) via its -p flag. This would definitely help you avoid a similar situation in the future, but I doubt a small employer would have the resources to properly implement this technique.
How you should proceed
Your SysAdmin is suspect in this case. Its highly unusual for him to need to update your machine locally, as updates can almost always be pushed centrally from a server to each initial client authenticated against it. There should be logs, if they have not been tampered with showing an user trail of actions taken.
If there a another person with admin privileges who you trust and you reasonably believe is not involved, explain what you saw, and only stick to facts, such as the IT admin frequently logging on your machine for "updates". Work with this person to gather some evidence such as logs from your email server. Depending on what you find, present to higher management or HR your facts.