If you're just sending mail to yourself, then use the sendToAdmins method of the MailService as that has the highest quotas: https://developers.google.com/appengine/docs/java/javadoc/com/google/appengine/api/mail/MailService#sendToAdmins(com.google.appengine.api.mail.MailService.Message)
By the way, note that your App is getting a Sockets exception. That means that you aren't actually using the App Engine Mail Service, almost certainly because you aren't using the JavaMail classes that are included in the runtime.
In any case, I recommend switching to the low level Mail API and using sendToAdmins. But if you don't do that, try using the included JavaMail classes instead of uploading them with your App.