But as far as I know, do we know for certain that Hawkwing had a hawk sigil on his scabbard? I can easily see Hawkwing carrying a sword with a dragon on it, merely because of the huge symbolical and historical meaning of it.


"As for Rand saying it reminded him of the last time he was in Falme, well, he ended up appearing in the sky, proclaiming for all the world, that the Dragon had been reborn, and, he used a similar sword to fight Ishamael with. (Those are two reasons right there.)"

I am not sure if Rand's old sword would remind him of this one other than they are both power-wrought. Tam's old sword also had a heron on it, and Rand's new one doesn't. Why wouldn't Rand think of his other-power wrought-sword that Aviendha gave him then?

Finally, Rand specifically mentioned that this was centuries old not thousands of years old. That timeline corresponds better to Justice than the alternative you speak of.

I think if we add in the prophecy that mentions the sword and Justice together, we have extra weight that the sword is actually Justice.