After rereading that section, it looks like he's trying to say that it's a sword we should recognize, which would seem to imply AH's sword Justice. We really don't have much of a description of it, as others have pointed out. It says it is "his great sword, Justice". I would note that it does not say "Great-Sword" (capitalized and hyphenated) or "great-sword" (hyphenated). In that case, it should probably be interpreted to mean it was "Justice, his famous and renowned sword".

So, that it "feels right" to Rand's hand, could be saying that as the DR, it is fitting that he is dispensing the "Creator's justice", with the sword, Justice. And, of course Rand would recognize it because he has seen it before at Falme.

The incongruity that confuses the issue would be the scabbard with the Dragon on it, which led me to my original conclusion that it was LTT's sword. However, we have already seen Rand have someone make him a replacement scabbard for Laman's blade, so we could assume the scabbard is not the original.