You should be able to determine the type of AA award by the AA ff miles needed. (compare to award chart)
AA now have saaver, aanytime, dynamic, married segment & web special awards.
And different ways of displaying. It is easier on laptop compared to a phone app.
Looking for awards on the QF web site may/may not work.
Example:
https://www.flyertalk.com/forum/amer...-ticketed.html
Edit
Looking on QF my ff account I can see(book?) LHR DFW 26 Dec BA193 15:45 20:10 1 off business class award (did not look for 2)
Adding PSP nothing shows. (Not that surprising for that destination over the holidays)
Do flights for cash show as being available on your dates?