The flight won't necessarily be full. If you're not logged in as an elite AAdvantage member, you won't see the "preferred plus" (purple) seats that AA is holding for elites. AA also blocks some seats for airport assignment, generally the rows in the very front (for the physically challenged with mobility issues) and very back (unofficial crew rest seat on light flights); I believe they'll also give these to families with young children who weren't able to get adjacent seat assignments.

If you provide your flight info, someone with status may be willing to log in and tell you how many preferred plus seats are available. Even with this, though, determining loads by seat maps is far from exact though because you don't know how many pax are holding confirmed tickets without seat assignments and which seats are blocked from pre-assignment in the first place. Sometimes doing a dummy booking yields more info; if AA.com won't sell you 6 seats, the flight is likely full, close to full, or overbooked.