Kataang's relationship status in the Aang film
The film takes place when Aang is in his mid-twenties - it's over a decade after the events of ATLA, so a long time where Aang and Katara have been together as a couple. Given that the show itself, and several of the comics, had Aang and Katara considering a future with marriage, it's interesting that they.... don't seem to be married at the beginning of the Aang film.
While Aang is already well into (young) adulthood, I firmly believe the film marks Aang's transition into well-established adulthood. It is a story about him letting go of his childhood (trauma) and being ready to settle down.
So why aren't Kataang married?
The film starts with Aang, alone, in the ruins of his childhood home. According to Katara's dialogue, he spends a lot of time there by himself (well, with Momo and Appa) and seems to be dwelling on his past a lot. He feels immense guilt about his people that he couldn't save.
On the other hand, Katara seems to live permanently in her own apartment in Republic City. Even the crew calls it "Katara's apartment" and the design makes it look like Katara's personal space.
However, Aang also visits there. It does feel like it's directly suggested they do have a more physically intimate relationship (more than anything we've seen from their teen romance). Aang stays at Katara's place as a casual thing, Katara waited up for him, he came back and kissed her goodbye before leaving... so it feels implied he does stay with Katara whenever he is in Republic City, and also, that Aang doesn't have an actual home of his own yet. Republic City is for all the four nations.... except Aang's.
It's interesting if Aang and Katara are not married and do not officially share a home, they still casually live together and, not to mention, carry symbolic relics of each other at all times. With Aang's beads on his pendant and Katara carrying yellow bands. Why would they do that if they aren't married? In the film, it feels like there is some higher level, more adult recognition between them of their relationship, beyond "boyfriend and girlfriend", even if not marriage. There could be several explanations, but my own current take on that is here. (I'd like to imagine they get married in both a nomadic sense and later a watertribe official marriage).
The whole Aang being shirtless scene felt quite intimate and a bit suggestive despite Aang's mood.
Katara talks about how the world needs Aang, but there's the unspoken echo here of "I need you". Aang is emotionally distant, and Katara wishes he was with her more.
Of course, Aang's isolation is not about him doubting his romantic commitment to Katara. It's that the emotional burden he carries from the loss of his people makes him distance himself from the present. At this point in his life, his trauma is particularly difficult, maybe exactly because he is at an age of really figuring out where he wants to settle down. It feels like this nomad boy, whose former home was completely destroyed, was not yet ready to take up a new one. If nothing else, from the guilt of knowing that no one else would hold onto it if he doesn't.
The whole emotional journey in this story is one of taking Aang from being emotionally distant by dwelling on the past, to following in blind faith a misguided attempt to bring back what he was holding onto. This almost caused him to lose everything he has (Katara and his friends, the peace between nations that he built), which would never make it worth it for him. In the end, Aang lets go of his guilt and realizes he already has what he needs to build his home with Katara, who reminds him that his culture still has a place in the present.
After a timeskip at the end, we see that the temple at Air Temple Island is being built.
Even if not stated clearly in the story, it feels quite likely that Aang and Katara got married or are just about to get married. Aang and Katara are about to move in together, officially, at a place they can both call home (an air temple on an island surrounded by water!), and Republic City is finally a place that contains Aang's culture. More importantly, his heart is ready to find a permanent home: Aang has stopped dwelling on the past, and instead, is dedicated to rebuilding his culture by sharing it's values and creating new things, not to mention, connecting with his loved ones.
Aang won't let his guilt hold him back from Katara anymore. It's not necessary to state if they're married or not, because we know exactly what Air Temple Island symbolizes - the life they shared together for decades and the family they created.
