You can only book MileSAAver awards on AA with Avios. You need to book this through BA either through the BA website or BA phone (which will involve a service fee).

However, you can search for MileSAAver awards here: https://www.aa.com/reservation/award...horEvent=false

You cannot book AAnytime awards, these are only available to AA members using AA miles and are available on any flight that has any seats available, hence why you see almost unlimited availability.