Filtering on Archive of Our Own is your main move, but it takes some finesse. Start by going to the 'Character' tag for Jax (I assume Mortal Kombat? 'Jax Briggs'). Once you're in his tag page, use the 'Additional Tags' field in the filters. Typing 'Dark' will give you options like 'Dark Themes', 'Dark Content', maybe 'Psychological Horror'. That's your foundation. Then, to ensure it's Reader-insert, you'll need to include 'Reader' and probably 'Reader-Insert' in the 'Relationships' filter. Honestly, I've found that pairing a dark content tag with the 'Dead Dove: Do Not Eat' warning usually gets the most intense, unflinching stuff—authors use that label when they really mean it.
You can also try excluding tags you know you don't want, like 'Fluff' or 'Happy Ending', to narrow things down. The search on that site is incredibly powerful but not always intuitive; sometimes you have to play with the 'Sort by' function. I'll filter by kudos or comments to find the ones that resonated with other readers, or sort by date updated to see new, active works. It's a bit of a process, but when you find a story that nails that specific blend of character voice and grim atmosphere, it's totally worth the effort. The comment sections on those fics often have great recommendations too.