EÉcrire à la main

En ce moment, j'écris rarement à la main -, (je le fais) seulement sur mon iPad dans mes cours magistraux, et encore, j'écris que quelques mots.

Hyphens are almost never used to introduce a complement to a sentence in French. Parentheses, (semi)colons or commas (depending on the context) are used instead.
"je N'écris que quelques mots" would be more formal (this would be alright in such a text, but somewhat colloquial in a formal letter or in an exam)

Souvent, je souligne des mots importants ou je dessine des diagrammes pour compléter les notes que mon prof m'a données, car je suis dans un programme scientifique.

The past participle ("donné") agrees with the direct object ("notes", feminine-plural) since it comes before the auxiliary verb "avoir" => les notes que mon prof m'a donnÉES

Si je devais écrire [un papier ou essaiarticle/une dissertation ?], je le ferais digitalnumériquement avec Microsoft Word.

"un essai" in the French-speaking literary tradition is typically a book where an author gives their opinion about a cultural or political matter. It's different from an "essay" in the Anglo-Saxon sense. What would come close for us would be "une dissertation" (a question involving a plan in several parts with logically ordered points/ideas. The best option is to keep the English word IMO, as it's something culturally specific.

If you mean "a paper" in the scientific sense of the term, it would be "un article"

Something "digital" is related to fingers, in French. Its use in the English sense is an anglicism, since the French word theoretically is "numérique"

Ça serait beaucoup plus pratique parce que mes cours attendent des submissionexige des devoirs remis en ligne et pas des copies papier (remises en main propre).

"attendre" is very rarely used with such abstract, inanimate subjects; "exiger" is though
"A submission" technically is "une soumission", but since the word is very polysemic, it's better to use a verb
The final part translates "in-person"

Autre que çaSinon/Autrement, je pense que je tape beaucoup plus vite que je peut écrire à la main'écris plus vite à la main qu'au clavier, et je le trouve ça moins fatiguant pour mes mains.

Other than this = sinon, autrement

It would be far better (syntactically simpler and more elegant) to have one general verb for both ("écrire") and different complements ("à la main" vs "au clavier")

You have a very abstract subject at the end which is a verbal phrase ("écrire à la main"), so it can be repeated pretty much only by "ceci/cela/ça" (although there are a few exceptions)

Je suis seulement coincé à écrire à la main pendant mes examens, mais j'écris pas des [essaisys/dissertations/articles ?] dans ce(s) cas - que-là, seulement quelques phrases.

You can pluralize "cas", since you had several exams already