While I do use the AA app (and apps from airlines I fly), I get AA text alerts, as well as text alerts from TripIt, I still glance at the airport monitors, just in case. Even when rushed, I still quickly glance at the monitor to verify I'm going to the right place. If I'm in an airport lounge, I'll look at monitors there (if available) and/or ask the lounge agent (and usually ask them how long it will take to the gate).

As for what to do, well you're pretty much at the mercy of AA and the agents. I've missed flights and now I usually plan things to pad my time quite a bit, just in case something happens along the way. When I've missed flights, I just pled my case...and was thankfully treated mercifully and put on the next flight (but I had status which definitely helped).