aa.com shows you the flights it wants you to buy, and surprisingly it shows different flights to corporate travel agents (last three trips I see no availability on aa.com for flights my TA can book and they see no availability for flights I can book!!).
Google flights seems to access a different set of flights again, as it now often shows me flights that aa.com won't, even though aa.com will let me book them if I follow the link from google flights (but not if I try to go back and look for alternatives on aa.com).
These problems seem particularly acute when trying to book Premium Economy. You'd almost think AA don't want to sell their empty seats.
So, the answer to your question is to dig around on google flights and see if you can find what you want there and then follow the link too aa.com.