I have a Send Email action set to After Record Submit. It has a condition which includes User != <my name> because I don't it to run for me. It's still sending the email when I cause the Workflow to run by submitting the record. Any ideas why?

Do you have an Admin role in your user access? Try to see Setup>Company>Email Preferences. There's a checkbox there Send to All Administrators. Not sure if that's what's causing it try to have a look at that page.