You don't have to call them. Cancel the trip online. Then go to https://prefunds.aa.com/refunds/, click "Continue to Refunds", enter your ticket number and name. At that point it will either tell you your ticket is eligible for a refund, and you'll give them some contact information, or they will say it is not eligible for a refund. If the latter, it will still let you proceed. It will ask for a reason, and you select involuntary cancellation or schedule change, note in the comment box which specific segments were cancelled or disappeared from your reservation, and give them contact information. Either way it goes into the system and will be in a "Pending Review" status for some days or weeks but you should get your refund.

I don't think there is any clear evidence that calling and requesting the refund gets you your money any faster.