Missed a deadline for a fellowship! I prepared a doctoral fellowship application package two months ago but I missed the deadline by a week because I confused the deadline for actual application with the deadline for recommendation letters. My advisors have sent the letters but I missed the application deadline. So I have made a huge mistake and I feel horrible! I am not an irresponsible person so I am in shock and have no idea how to handle the situation.It's the first time I am making a mistake like this and I feel so humiliated. I want to avoid my advisors for a while because I know they've put a lot of work into writing letters for me. What would you recommend? Should I call the research institution and ask if they could make an exception? Or should I just email the package and wait for their response? What should I tell to my professors(If they find out, I will feel humiliated!)?

Thanks a lot!
I would mail the application and contact the institution to explain the situation. If they are willing to make an exception, it's better if they receive your application as soon as possible, and if not they'll just discard it anyway.

Unless, of course, it is explicitly stated somewhere that late applications will not be considered. In that case there's nothing you can do.