I don't know about American, but I think most (certain for Frontier) will not charge anything for checking stroller or car seat. This is true even if you check them at the ticket counter. Most people check them at the gate, as you really don't want to lose them.

Just looked on AA.com. try this website

Strollers are exempt from checked baggage fees.

Highly recommend you ask about extra empty seats when you arrive at the ticket counter/gate.  About 1/2 of the time we have luck that there is an empty seat on the aircraft.  They can rearrange some seating assignments a bit to get you next to the empty seat.  Then you can have the kid in the car seat (if the carseat is aircraft approved).  Depending on the age of your kid, that could make things much less stressful.  You quite often have to move to the very rear of the plane (where there are more unbooked seats), but it is worth it.