Am I reading that you don't have any status with AA or any other OW carrier?
Yes, AA does have a policy for kids (I am not sure what the age cut-off is, but I think it is 11) that they won't be assigned seats by themselves. I mean, there are some blocked seats to help the airline deal with this issue.
If I am to guess what would happen if you don't buy a seat, you will get assigned 2 and 2 in separate parts of the plane. I don't think the price of the paid seats will increase, so you can keep checking as people change their plans to see if you find better, free seats. Checking once a day is not a problem.
I assume you have the 10% infant ticket, so you may be able to call-in to put your request for the bassinet areas of the plane.