I've found the best Katara/Zuko stuff on Archive of Our Own, but that's only if you're willing to dig through some serious tagging. Their filtering system saves lives. You can sort by kudos or bookmarks to avoid the stuff that's just... there. It’s not all gold, obviously, but the writers who care about character voices and canonical nuance tend to gather there, especially for Zutara. You get these epic slow-burn fics that rebuild their post-war relationship from the ground up, which is exactly what I want. The comment culture is also a lot more constructive, so the good writers stick around and polish their work.
Sometimes I peek back at FanFiction.net out of nostalgia, and there are definitely some classics from the mid-2000s fandom still kicking around. 'Embers' by Vathara, which is more of a worldbuilding epic with Zutara as a slow-developing subplot, started there. The quality can be a real mixed bag, though, and the site itself feels clunky now. Searching by favorites is your best move. A smaller, dedicated spot is The Southern Water Tribe, a forum that was huge for Zutara back in the day. It's quiet now, but the archives are a treasure trove of pre-'Legend of Korra' speculation and character studies that feel very pure to that era of the fandom. It’s almost like a time capsule.