Azula is one of the hardest characters to ship realistically in Avatar, due to her emotional state. But there are three that I believe will work and have solid foundations for, let's go through them, shall we? Ordered from 3 to 1.
3. Azutara (Azula x Katara)
*Artist credit to AlavastirCloud*
Now, this will take work, but they have similarities, they both have mother-related trauma, both are prodigies from their respective nations. Katara and Azula also have opposing personalities that would mesh well once they both get past their growth. Fire and Ice? Yes, please!
They have been few people that have been unphased or not intimidated by Azula and Mai is one of them. They have years of history and shared upbringing by Fire Nation ideology. Mai's dry sarcasm and emotional numbness helps with Azula's intensity and balances her out, same with Azula to Mai. Once reconciliation happens (Avatar: Cookbook, Azula Lighting, check it out) its very believable that Mai can work through Azula's shortcomings and also Boiling Rock adds layers when Mai said "I love Zuko more than I fear you" Speaking of Zuko...
Isn't it obvious?, my OTP for Azula is Zuko. Azula literally went out of her way to bring her bro back to the Fire Nation in Crossroads of Destiny and when she said "I need you, Zuko" in an uncharacteristically soft tone and not to mention, while the post-ATLA comics is a lot to be desired, they get their dynamics right and Zuko via his change, changed Azula for the better and her appearance in "Smoke and Shadow" proves it. Azula literally let's the mask slip around him (Ba Sing Se, multiple instances in early Season 3) If you want more depth into their relationship and subtle incest check out @hello-nichya-here and @icewhisker21 posts for more context
Thanks for coming to my TEDtalk!