How to refer to a university that changed its name?
Always use the name of the school that was correct at the time when you were doing something. So, if you mention your first degree, use University A and when you mention your current activities, use University B. If there is a chance that whatever you are writing is read by somebody from outside this university or who joins this university years later and could be oblivious of its past name, use footnotes or parentheses to indicate the name change like this:

University A School of Engineering (now University B School of Engineering)
University B School of Engineering (formely University A School of Engineering)

Furthermore use pronouns and referrers to avoid confusion, when both of them appear close-by, e.g.:

I apply to University B School of Engineering because I already did my BSc at this institution and this was a positive experience.

Unless the university failed to make all its personell and students aware of the change, I do not see a reason to mention the name change on documents that do not leave this university like essays and applications. (In particular the administrative staff is probably rather annoyed by the name change and prefers to be reminded of it as little as possible.)