OP - Sometimes the terminology throws off low-level employees. Call AA back, ask the same question again. Do not mention that you have already called and simply ask about making the change. If you get the same answer, ask where you can find the "fare rules".

In the meantime, if you would please provide the following information, people here might be able to help you:
1. In what country did you purchase your ticket?
2. Origin?
3. Destination?
4. What is the fare basis? (This will be a five-digit code following the origin and destination codes). For instance, if you are flying JFK-LHR, there will be a long list of what looks like gibberish at the bottom of your e-ticket receipt which will show "JFK-LHR XXXXX" where the 'XXXXX' is the fare basis.

There should also be language in the e-ticket receipt about whether and how the e-ticket can be changed.

Lastly, as others note, it may be easiest to call your travel insurer to determine how best to handle this. It may be that the insurance company will simply cover a new ticket.