Another theme is redemption. Sometimes, the fanfics explore the idea of Azula's possible redemption. Aang, being the symbol of peace and balance, might be the catalyst for her change. Their interactions could lead to Azula re - evaluating her actions and beliefs. This theme adds depth to their relationship in the fanfics.
You can try some popular fanfiction websites like Archive of Our Own (AO3). It has a large collection of fanfics, and you might be able to find some great 'azula vs aang' ones there. Just use the search function on the site.
In some fanfictions, Aang's connection to his past lives plays a crucial role. He could draw on the wisdom and skills of previous Avatars. They might guide him on how to defeat Azula specifically. He could learn a new bending technique from one of his past lives that gives him an edge over Azula's firebending, like a unique way of using water to extinguish her most powerful fire blasts.
There could be a fanfic about their alliance. Maybe in an alternate universe where they have to team up against a common enemy. They could start off as enemies but gradually learn to respect and rely on each other.
Their relationship could deepen as they share in the joys and sorrows of rebuilding the world. They've been through a lot during the war, and now they can truly rely on each other. They might become even closer friends at first, with a mutual respect for each other's abilities growing stronger.
In some cases, the anger could be a catalyst for a more romantic development. Aang expressing his strong emotions might make Katara realize the depth of his feelings for her. Or it could be the other way around, Katara's reaction to his anger could make Aang see her in a new light, leading to a shift in their relationship dynamic towards a more romantic one.
There's also 'Azula's Redemption with Katara'. This fanfic focuses on Azula's journey towards redemption with Katara as a guiding force. It shows how Katara's compassion can have an impact on someone like Azula, and it has some really touching moments.
Katara could be heartbroken. She might initially be in denial, not believing that Aang has actually left. Then she could become angry, feeling betrayed by Aang's sudden departure.