I agree with you about Another, even it had me cringing, definitely overdone. I think that one classroom scene has forever left its mark in people's memories.
Death Parade is kind of like Another and a bit of Jigoku Shoujo, only its take on the genre is very interesting. I think it's something you have to watch, but it does have gross, gritty, or disturbing moments.
As for Naoki Urasawa's Monster, it's hard to explain. But it's very gritty. It's honestly one of my favourites along with Death Parade. But I liked Monster more than Death Parade, for sure.
Death Parade
en.wikipedia.org/wiki/Death_Pa…Naoki Urasawa's Monster
en.wikipedia.org/wiki/Monster_…