Ok, to follow-up. I just looked at my SE profile, and it now says that I have FFXIV as my one registered game. Perhaps on one of those "There was a problem. Please try again later." messages, it actually went through, and just psyched me out.
This is most-likely the cause of the "There was a problem with your code. Please make sure you have entered it correctly." error.
For anyone that has entered your 18-digit code and gotten the error message, go check your SE profile and see if you have FFXIV registered to your account.
Edited, Aug 20th 2013 10:06am by Teracide
Not sure if anyone answered you yet, but that is probably just registered as beta and not full version, especially since I'm assuming you never even entered your registration code. I'd say have some patience. I'm sure server will die down long enough to register between now and Saturday. Good luck :)