Whenever I make telephone reservations, the last thing before the "Is there anything else I can help you with today?" part is where they read back to me from the screen the reservation I made. It is likely that you weren't paying close attention, assuming that the reservation was what you wanted.

In addition, a red flag should have been that they were putting you on flights that you had already researched and knew there was no space on.

AA really doesn't even have to allow you to cancel the flight. Fortunately for you, they are allowing that. But they are not responsible for what you did prior to receiving their confirming e-mail.

Since AA has no seats at all to St. Kitts for the dates you want to fly and will cancel your ticket, I would check availability with other airlines.