Just to clarify - this is the response I got from their Admissions Director when I asked about the timeline and if I have to submit by R1 deadline.
"You do not need to submit your fellowship essays by Oct. 6. Plan to have it submitted here by Nov. 30 to be considered with other Round 1 applicants.
You may apply to any fellowship you are interested in. Applying to a fellowship and fellowship essays have no influence on the overall admissions process at Goizueta. Center directors select their own fellows after a student has been slated for admission."