Hello, my name is Judith. Thank you for this opportunity to answer your questions.
Your DSO did tell you the only option you have available and you would have to set forth why your application was sent 5 days beyond the allowable 30 days. The reason has to be that there were some circumstances beyond your control, such as a medical emergency for you or a close family member, etc.
I do note in your outline above that you said your graduation date was 8/16 but your Diploma says 9/7/13. What is controlling is the date on your I-20 which says the completion date when you finished your requirements. The Diploma would have the Commencement Date when the Diploma was awarded but often that is after you have met the degree requirements and completed your course of study.
So you would not have an argument that you should be able to file a new I-765 up until 11/7/13 if your I-20 has an earlier completion date.
This is most unfortunate. Regrettably, it happens often, students do not know of the 30 day deadline, but there is no exception in the law for lack of knowledge.
You only other option now would be to enroll in graduate studies and seek a reinstatement of your F1 status. You might be able to qualify for Curriculum Practical Training if the advisor feels that it would be an integral part of your program of study.