Native English speaker here - my English teacher at school always used to say that in order to write well, you need to read well.

That probably doesn’t help you in the short term, however! But, yes - finding “good” authors you enjoy is a good way to go.

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.

I don't think anyone will be able to give you general tips that actually help. You should find someone willing to edit you. That's the best way to improve at writing.

I don't know if there's a source specifically for this but when I often get stuck and start repeating myself I look up Merriam Webster's thesaurus and start replacing boring words with better ones.

So rather than say something like awesome I'd say a word like marvelous, lovely, etc.

Career fairs and some colleges may have writing centers or people that assist with cover letters, resumes, etc.

If you dont have that opportunity, I would search some up. Find ones that resonate with you and copy it but make it your own. Thats what I have done and ive been fairly successful. You can tie personal stories in them and still make them sound professional.

Read lots of literature. Find fiction that you enjoy and read it every day.

FWIW I think you write well. I would've assumed you were a native speaker.