I think your expectations are a little low, especially for a refund of a domestic ticket. The Customer Commitment negotiations of the early 2000s with federal regulators established deadlines for refund processing. You were clearly entitled to a refund, by DOT guidance: AA couldn't offer you another non-stop (operated by AA) within a modest time change of your original ticketed flight.

For all eligible paper and electronic tickets and optional products and services purchased within the U.S. with a credit card or cash, refunds will be provided within 7 business days of receipt of the required refund information.

https://www.aa.com/i18n/customer-ser...rvice-plan.jsp

So, AA has a reasonably quick routine to do something they probably do a thousand times a day? That's way overdue, IMHO. AA just got slapped with a DOT fine for slow refund processing. https://www.reuters.com/article/us-u...-idUSKBN1A62OY

I'll point out that authorizing a check isn't the same as producing and mailing a check. Good luck.