Completely agree that you should not use letterhead from the academic institution you are trying to leave.
I put those letters at the bottom of the pile when reviewing job applications for teaching positions.
I don't think this advice is universally applicable; I know in many other disciplines, cover letters are read carefully, and thus their professionalism will have some salutary effect.
My personal feeling is that using your current institution's letterhead, your office address, etc.
Looking at examples will give you a better idea of what various letterheads look like and how to accomplish the design you’re after.
We've gathered some examples in this article, but you can also try searching for "cover letter design" on Pinterest for inspiration. You can buy letterhead templates made for Word from Etsy.
This practice extended to the English with houses large enough to have guest rooms, where the etiquette is to provide headed notepaper of your own for your guests to use. There are strong and conflicting opinions on this ("it would set off a red flag" to not use institutional letterhead, etc.
vs "I put those letters at the bottom of the pile.") We should ask ourselves, as academics: if we are evaluating people's job applications using such incredibly fine distinctions in academic etiquette, either: 1) we are prioritizing completely useless information and probably introducing a good deal of bias against folks like international students and first-gen college students along with it!
I have faith that people are not actually making such important decisions using trivia.
I therefore suggest that the original poster should choose whichever option allows them to make the content of their letter clearer, i.e., if you need the space for more information, don't use the letterhead!