If you want a good, thoughtful appraisal of your character and abilities, give your writer ample time to work on the letter. Committees want to see that your letter writers know you, in class and outside of class if possible. Selection committees rely heavily upon letters of recommendation. A letter that includes specifics about the letter writers interactions with and observations of you will be far more influential than one that just indicates that you've received good grades or repeats what you may have included on the application. It is important for you to cultivate associations with professors outside the classroom.
Deadlines are of paramount importance.
Materials submitted for campus deadlines should generally be as complete and professional as you would submit to an external review committee. The Fellowship Advisory Committee will review your application, interview you and give you feedback. Those applications which show promise and are deemed worthy of the award are then nominated to be forwarded to the next level. You will continue to make finishing touches to the application with the Fellowships Adviser and Faculty Representative until you feel it is ready to be mailed. For institutionally-nominated scholarships, you will turn in the final copy and all supporting materials to the Fellowships Office well before the deadline.