As someone with the opposite problem, too formal and not very good at casual writing. Truth is, formal writing is robotic and in today's context it is regarded as awkward except in a few places. Most of the samples online that the bots are trained with are overly formal examples. 99% of cover letters are never published online, so that's an area they're lacking. What they have access to is the awfully generic slop that's impersonal and meant to sell online workshops about writing cover letters.

There's a very difficult task in making formal writing feel natural and warm. I would advice instead to aim for transparency. A cover letter is supposed to highlight a match between your skills and personality, with the company role's needs and work culture. It's not a cold sales pitch, you must show that you did your due diligence about getting to know the place before applying for the job. As long as it sounds like the genuine you talking, not a façade, it doesn't has to be too formal, just keep the content and vocabulary professional. How you would talk in the workspace with a coworker that you don't know too well yet. A cover letter is more like corporate flirting than lawyer speak.

As for material, read the basic common sense guides online, but, and it is a big but. Also read a lot in general, specially in English as it isn't your first language. Unlike LLMs humans are actually intelligent and we can use experiences from other contexts, and good writing in general shares common principles across all genres. Even if every genre has specificities, they're usually an addendum or exception of general good writing. Variety is the spice of life.