For the record, I think both were handled horribly so let's not split hairs too much.
Fortuneteller is just foreshadow, there was no actual development between Aang and Katara.
Cave of Two Lovers is much the same. There was a vague blush that happened at the end that hinted maybe (maybe) Katara had feelings for Aang, but that wasn't developed on at all.
The Headband, same shit.
Invasion was Aang suddenly kissing Katara, which Katara later expressed confusion over. This was never resolved by the way.
Korra and Asami had very little in the way of romantic hints, but there were a lot of episodes in season 3 and 4 that show them becoming a lot more intimate. Aang and Katara's relationship isn't really developed at all in book 3. Either way there wasn't enough in the narrative to support them getting together in the end.