I have often driven pass University of Chicago on numerous occasions and thought about how I would fit in at the Pritzker School of Medicine, which has a reputation for being a school that has trained many world renowned physicians of different fields of medicine.
I assume you mean "driven past."
"Often" and "on numerous occasions" are redundant: choose one.
"In" not "of" different fields of medicine.
One mutual endeavor possessed by Pritzker and I, is the desire to serve the community.
A "mutual endeavor" is something that two or more people do together. You and Pritzker have not done anything together yet. Furthermore, one cannot "possess" an endeavor. Also, an endeavor is an activity while a desire is an emotion.
From just these two sentences, we can see that indeed this essay is very rough. You will need to significantly improve your the coherence of your prose if you hope to get into any medical school. Think about what you are saying. Look back at a sentence after you have written it and ask: Does this make sense?
Is it better to have less word or should I try to get as close to the word count stated as possible. 3500
In general, it is better to use as few words as necessary to make one's points. However, in a medical school application, you will want to say as much as possible. Many of the things that you refer to only briefly here -- such as working at the free clinic -- ought to be the subject of several detailed sentences.