And I've gotten I'll.

This sentence is fine, just some more information to you: in British English (or at least to me), this sentence sounds a bit unusual. We usually use "got" as the participle for "to get" rather than "gotten". I would personally express it as something like

"And I've become ill."

But I'm sure this is fine in American English, which I imagine is the style you are learning anyway. I understand that many learners appreciate me pointing out when there are differences between British and American English.

Autumn and winter are some kind ofrather sad and depressing (times).

"Bleak" is a good word to describe Autumn by the way.

Just to link this to a previous comment of mine where I pointed out a difference in American and British English, you've used the British name for the season here (as opposed to "Fall" in American English), but used more American sentence construction above.

And I've gotten I'ill.

This feels like a phone keyboard error :)

I used to love autumn but I don't like it now.

Would be a little more natural here to say "but I don't like it any more", I think.

Autumn and winter are some kind of sad and depressing time.

I think this is probably какие-то, and it's kind of difficult to translate :) This seems the most natural thing to say. Maybe a little colloquial. "Autumn and winter are sad and depressing in some ways", "There's something sad and depressing about autumn and winter", or just using adjectives for degree like "Autumn and winter are a little sad and depressing", "Autumn and winter are quite sad and depressing".

There's nothing very wrong with calling them "times", but it's not necessary here.