If you like webtoons, the online manhwa Noblesse is exactly what you’re looking for (slice of life + fantasy). You can read it legally in the Naver webtoon app.

I don’t know any magic trick to find fantasy or science fiction books that break the epic hero stereotype. I just brows through the books and pick one that seems interesting. Just by reading the back of the book I can usually discern if it follows those stereotypes or not, though sometimes the end result is still surprising. For example, Brandon Sanderson’s Mistborn series is basically a large stab at those stereotypes (both following and not following them).

Some recommendations:

• Ursula LeGuin’s books and short stories regularly have a very strong ‘slice of life feeling’ to me. It’s one of my favorite authors. If you’d like to read about normal life in a fantasy or science fiction setting, her short stories are the way to go.
• Robin Hobb: Her books do have a central hero (who doesn’t want to be a hero) and the issues are larger than life kinds of problems, but they don’t follow the LOTR epic quest stereotype at all and they’re amazing to read.

In my experience, female authors are a bit more likely to write outside those stereotypes, but that’s certainly not a general rule.

A lot of fantasy and science fiction authors tend to write about larger than life and/or world-threatening affairs, so if you don’t want that your options are going to be very limited within these genres. If you’re also happy with a book that has both large issues and a ‘slice of life’ feel, you’re going to have a lot more options. For example, there are a lot of books with a “school life” feel together with a larger revolving plot (the Harry Potter books, Trudi Canavan’s Black Magician trilogy, …). I would also recommend checking out Garth Nix’s books (especially the books of the old kingdom), because he also shows what it’s like to live in the fantasy worlds he creates (especially in the book Lirael, though that’s already the second book in a series).