At least at my airport (STL), AA is super-strict about the time-deadline for checking in bags. Check the posted deadline the day before you leave (it can change on you), and whatever you do, DON'T MISS THE DEADLINE!
If they say that the bags must be checked one hour before the scheduled flight time, be sure that you get them checked by that deadline. We were at the airport an hour and 45 minutes ahead of time, but we ended up in a long bag-check line, and we missed the deadline by 2 minutes (yes, you read that correctly, two minutes.) They refused to let us on the flight. At that time, they were charging to let people fly standby, so those two minutes cost us nearly $500 in cash. I think they have backed down on that charge now, but they are still being really strict about that bag deadline.
If you are at all running late, drop the necessary people off to get the bags checked first thing, then worry about the car, etc.