Let's look at the following quote:

"He fingered the cloth-tied hilt. The weapon was long, slightly curved, and the lacquered scabbard was painted with a long, sinuous dragon of red and gold. It looked as if it had been designed specifically for Rand — and yet it was centuries old, unearthed only recently. How odd, that they should find this now, he thought, and make a gift of it to me, completely unaware of what they were holding..."

Rand notes the "sinuous dragon of red and gold", and then thinks "It looked as if it had been designed specifically for Rand - and yet it was centuries old".

The two "its" in the "It looked as if it had been designed for Rand" quote must refer to the scabbard (or to the scabbard/sword combination), since it is the scabbard that displays a symbol related to Rand, not the sword itself. There is nothing about the sword alone, that we know of, that would make Rand think that it alone had been designed for Rand (IIRC).

So, when Rand thinks "yet it was centuries old", that "it" must again refer to the scabbard (or scabbard/sword combo). Meaning, the scabbard and sword are both centuries old, not just the sword.

The scabbard is lacquered, and while I'm no expert on the topic, it seems possible that the lacquer prevented the scabbard (assuming it is made of leather, as you speculated) from deteriorating while under water. Either that, or the sword/scabbard combo was encased in a watertight container.

Either way, I feel pretty confident that the scabbard is as old as the sword, based on the quote. In which case, why the scabbard has a red & gold dragon on it is a good question.

Also, AH was originally from Shandalle, which is now Cairhien. That was his seat of power, from what I can gather, until he started (very late in his life) on the new "captital city" in that stedding, where the statue of him is. It's said that AH died on the day the statue was completed, so I would guess that he was still in Shandalle at the time, since at that point only the statue had been completed (it was the first thing begun, and took 5 years to complete).

Therefore, I think Cairhien is a good guess for where the sword was found. Probably not under the CK - as you noted, that idea is problematic. But perhaps a different statue. We know there are a lot of scholars gathered in Cairhien re Rand's academy there, so that would fit as well.

It seems possible the AH was simply buried in what is now Cairhien, along with his sword. And centuries later the cemetary/tomb/etc. had disappeared, and a statue (unrelated to AH) had been arbitrarily erected over that site. The fact that the statue was partially submerged could simply be due to weather - winter snow melt, etc.

It's a boring explanation, I admit. But it seems plausible.