I suggest using Skype to call the US number directly*, as the US call centre folks actually know what they're doing. Skype is always free for US toll free numbers (in addition to UK ones, for what it's worth), and I find the people answering the UK call centre number for AAdvantage Customer Service are utterly useless. If you haven't got broadband at home, try calling from work, from a coffee shop, from a hotel next time you're on a business trip; even your local may have WiFi now, depending on what part of the UK you live.

* Please note that I mean "directly", not "dreckly".