Contractions are colloquial. I recall reading in a few places and hearing from a few teachers early on, contractions should be avoided. Some English teachers enforce it as a strict rule while others don't care.
Outside of English, as I've found, professors will throw sacred dictates such as bibliographies clear out the window. Whatever the case, if you're ordered not to use them, don't. If you're writing a formal essay (including most admissions statements) and unsure, don't use them. If you're writing a fairly academic paper, as you are required to do often in certain fields, definitely avoid them. Contractions functionally disappear as the topic of interest becomes complex and technical.
The common sense way to avoid a word is by writing what you mean to say in a different way. This ability can be developed through mastery of the language.
Take your first sentence for example. It can be adapted to exclude any word without losing vital meaning.
"I discovered that I didn't tremble anymore when I had to speak publicly."
It was relieving to find that my body no longer trembled while speaking publicly.
I realized that I didn't tremble anymore when I had to speak publicly.
The trembling which I had reluctantly accepted as an incurable symptom of public speaking, subsided, and shortly thereafter disappeared completely.
Here is a sentence that avoids all of what was contained in your original sentence, while staying close in meaning:
My body no longer shook while addressing an assembly of people, it occurred in the back of my mind.
If you can't (find yourself unable to) (are incapable of acting to) (lack the ability to) (do not possess the skill to) (try and fail at base, to) (beg the skill to) etc. etc. etc. paraphrase a sentence in a half dozen or more ways, your language skills may need serious work.
Mustafa