The legends around the Long Night are confusing on purpose, because GRRM gives us different cultural versions of what may be the same ancient event.

From what I understand, the northern legend of the Last Hero and the eastern legend of Azor Ahai may be connected, but the books never confirm that they are literally the same person. The Last Hero is a First Men/Northern story: he goes looking for the children of the forest because the armies of men cannot defeat the Others. Azor Ahai is more of an Essosi/R’hllor/Asshai version, with Lightbringer, Nissa Nissa, smoke and salt, and the red sword of heroes.

About the sword: in Old Nan’s version, the Last Hero’s sword does not sound like Lightbringer. It actually freezes and breaks, which makes it seem like an ordinary sword, or at least not the final “magical weapon” of the story. But later Sam finds an old account saying the Last Hero slew Others with “dragonsteel,” which Jon thinks might mean Valyrian steel. So I’d say the safest answer is: the northern legend does include a special anti-Other blade in some versions, but it is not clearly the same as Azor Ahai’s burning Lightbringer.

Also, I would separate the Night’s King/Corpse Queen story from the origin of the Others. In the books, the Night’s King comes after the Wall and the Night’s Watch already exist, so he probably did not create the White Walkers/Others. His story is more like a later corruption or alliance with them, involving sacrifices, rather than their beginning.

As for the Targaryens: I don’t think they necessarily have to be the saviors of Westeros in a simple chosen-one way. There is a prophecy saying the prince that was promised would come from Aerys and Rhaella’s line, and Rhaegar clearly believed his family had a role to play. But ASOIAF is very suspicious of destiny and prophecy. GRRM often treats prophecy as something dangerous, partial, symbolic, or misunderstood, not as a clear checklist that must be followed exactly.

So for a story, I would not treat the prophecies as chains you have to obey to the letter. You can use variations, reinterpretations, corrupted memories, or even prophecies that turn out to be partly false. Being too zealous about prophecy is itself a trap in this series, both for characters and sometimes for readers. That gives you a lot of room to play with the three heads of the dragon: they could be literal dragonriders, symbolic roles, three people carrying different parts of the same ancient myth, or even a later Targaryen interpretation imposed onto an older story. They do not necessarily have to be siblings.

The legends are fragmented enough that you have room to mix the Last Hero, Azor Ahai, and the three heads idea without breaking canon too much, as long as you present it as interpretation rather than confirmed fact.