This is really trivial, but it is acceptable to just refer to the university as “Hopkins” in the essay? Writing the fully name Johns Hopkins just feels like it disrupts the flow of the paper.
JHU uses the term “Hopkins” themselves on parts of their website, so if it helps the flow of the paper, I don’t see why you can’t use it.
Example:
"We invite you to read through this website to become familiar with the many ways to be engaged and involved as a Hopkins parent, and we encourage you to visit our Class of 2019 Parent page. "
Source: http://parents.jhu.edu/
Referring to JHU as Hopkins is absolutely acceptable, and sometimes even more natural than JHU or Johns Hopkins!