Usually there is no real difference between submitting your essay weeks in advance and sending it by express mail on the last day. An office assistant will put your essay in the pile for your readers, all the same. The advantage of submitting your application early is that if there is a problem, you might be contacted early enough to be able to correct it. The advantage of taking extra time is that you might need it to improve your essay.
If it looks like you are going to miss the deadline, it's worth a shot finding out if you can submit part of your application by the deadline and then the essay a day or two later. These exceptions are made for letters of recommendation all the time, because the applicant should not have to suffer from someone else's mistake. You never know if the office assistant is going to be in a good mood and might generously let you sneak by, even when the mistake is your own. Make direct contact with the organization if you can, and see what you can do.