For documentation sake, I wanted to post what fixed the problem. Following @trent's recommendation and the documentation, we changed the Resource to be

<Resource name="mail/Session" auth="Container"
       type="javax.mail.Session"
       mail.debug="false"
       mail.smtp.host="smtp.washington.edu"
       mail.smtp.user="xxxxx"
       mail.smtp.password="xxxxx"
       mail.smtp.port="465"
       mail.smtp.auth="true"
       mail.smtp.starttls.enable="true"
       mail.smtp.socketFactory.class="javax.net.ssl.SSLSocketFactory"
       description="email resource"/>

After a restart of the LabKey Server, email was working.