Then I checked Partner Airlines. They have plenty of award tickets available for their own mileage customers, unfortunately these same award tickets are not available to BA miles holders.
There are known issues with ba.com that may be affecting you, for example it is not generally possible to book an AA award ticket on ba.com that requires a connection. If the issue is that you can see sAAver awards on aa.com but these do not show up on ba.com, then you can almost certainly book these by calling the BA reservations number. If you tell them the problems you're having they also ought to waive the telephone reservation fee. Note that you cannot get AA's AAnytime awards using BA miles.