Hey everybody! I did a little more sleuthing and managed to solve the problem.
First, I went to the Square Enix support page here.
Second, I clicked on the link for FF XIV ARR, which took me here.
Third, in the search field I typed in registration code, which brought me here.
Then, out of curiosity, I clicked the link that said "When does Early Access end?" That brought up this page, which not only answered my question, but solved this problem. You can see that page here, and follow it's instructions to register the actual game. I hope that helps everybody.

tl;dr click that last link

Holy shit, I forgot to tell y'all how I found the actual code. I went back to Amazon, went to My Games, and under ARR there was a button to view the product key. Click that and it will show you a different code than the one you used for Early Access. That's how it worked on Amazon. I don't know about other services.