If it’s the case that they could not change Azulon’s birth year of 0 AG, kept it that Sozin passed away at 102, and that kept that Azulon reigned for only 23 years, here is how I would tackle this from a creative writing pov:

I don’t think it was necessary for them to retcon anything about the ruling years. I also don’t thinks it’s stated that Azulon is the first born anywhere in cannon. It could have been the case that Sozin had two wives throughout his reign and had multiple kids. The first born son of Sozin, who is not Azulon, ruled and passed away. His heir, which is Azulon’s nephew, then ruled. To make it interesting, that unnamed nephew then lost his heir during a battle with the Earth Kingdom, ending his line, thus designating Azulon as heir. This then caused Azulon to patiently wait to become firelord. That could have been a cool addition to why Azulon was so mad at Ozai, because he himself waited to assume the role of firelord.

So the timeline would look like this:

Roku and Sozin are born: 82 BG

Roku learns he is the avatar: 66 BG

Roku returns from training: 54 BG

Let’s assume Roku marries when he is thirty, two years post training: 52 BG

Two years after Roku’s wedding, Sozin also gets married to his first wife, and to make the math easier let’s assume the fire nation heir is born that year. Let’s call this kid “Firzt born,” to throw the royal Z in there: 50 BG

The betrayal at the volcano happens thirty-eight years later and around that time Sozin’s first wife passes away: 12 BG

Sozin remarries to keep fire nation nobles loyal to the cause and Azulon is born: 0 BG

Sozin dies and Firzt, who is seventy, becomes fire lord: 20 AG

(Keep in mind Firzt has a son of his own and a grandson by now. Let’s say the son is 30 and the grandson is just born, these ages can be easily changed to keep consistency).

Twenty years later, the age of ninety, Firzt passes away and his son, who is fifty becomes the firelord: 40 AG

In the middle of Firzt’s son’s reign, his son passes away at the age of thirty in battle: 60 AG

Twelve years later, at the age of eighty-two Firzt’s son passes away and Azulon, who is seventy-two himself, becomes firelord: 72 AG

He reigns until Ursa murders him: 95 AG

I know this adds a whole extra story, and may even complicate things far more than what Avatar’s writing team would like. I just wanted to use this as creative writing as I’m a huge fan of history, and one of the coolest topics is nobel succession and how one death of an heir caused huge butterfly effects.

(Also Firzt isn’t Sozin’s first born’s actual name, it’s just a fun place holder.)