Is there a more politically correct/respectful word for maniac/unhinged I can use on a university application?

A few words you could use to describe your vast energy reserves:

Unbridled

Boundless

Inexhaustible

Bottomless

Indefatigable

Any of these convey the notion that you had a ton of energy to direct at some pursuit. Unbridled in particular fits well here, as it also connotes that your energy wasn't otherwise focused or used in a productive way.

In that context, you might want to go with irrepressible (or unbridled as suggested by Nuclear Wang).

Irrepressible (adj): full of energy and enthusiasm; impossible to stop.

Example: Even the rain failed to dampen his irrepressible spirits. [Cambridge English Dictionary]

You could also replace 'unhinged energy' with hyperactivity or tireless enthusiasm.

"Maniac" is not quite it, but manic fits well. Although "manic" has a psychiatry-specific connotation, it also has an informal role to roughly mean driven.

[Oxford Online] Manic: frenetically busy; frantic. "the pace is utterly manic" Similar: frenzied feverish frenetic

Drama was the ideal outlet for my manic energy and whacky humor.

Perhaps the word uncontainable doesn't connote 'out of control' too much nowadays.

uncontainable [adjective]:

(especially of an emotion) very strong.

• his uncontainable enthusiasm

[Lexico]