"Rand's sword isn't Justice, why would Hakwing's sword be found under rocks in the sea near Arad Doman, of all places? (and let's not go into the "He gave it to Luthair who accidentally dropped it on his way to Seanchan" theories). And why on earth would Hawkwing of all people choose a dragon for his scabbard, he who spent years hunting down the False Dragon that caused the most chaos in the Age? How would a sword and scabbard survive intact under water unless it's both from material that are very resistant... like made in the AOL resistant..."

Maybe an Aes Sedai had a foretelling that told her to put Hawkwing's sword there? Hawkwing could easily have had a power-wrought sword. A man of his status would almost certainly have had a power-wrought sword.

What chapter was it in that said it was found under-water? I have to reread that part again. Was it in chapter 1 that Tor put online for free?

I still don't understand why Rand would have remembered Lews Therin's sword from his own memories. Was there a scene where Rand saw Lews Therin's sword? In the Aiel ter'angreal?
Maybe this was an effect of Rand going insane and getting his own memories confused? This whole sword thing doesn't seem clear to me at all.