Thanks. I'm out of the office until Monday, but I'll try your suggestions then. I do remember seeing something about SMTP errors in the mattermost log.
EDIT: Actually I was able to see the logs via the system console, and this is the error I'm getting:
[2016/03/23 02:30:53 CDT] [EROR] SMTP server settings do not appear to be configured properly err= details=dial tcp: lookup mail.mydomain.com on [::1]:53: read udp [::1]:33043->[::1]:53: read: connection refused
EDIT 2: Okay, so looking through the logs, I'm seeing that I may have been using the wrong domain name for the email address... I'm an idiot Hopefully fixing that on Monday will get it working again.