Concerned about upcoming flight - website says seats are full?

You can check in online from 24 hours before departure. Check in as early as possible to get the best selections of seats to choose from. Even if you don't check in until last minute you will still get a seat, but it will be whatever seat is left over.

Yes, the airline may oversell the flight, but that is done by careful calculations of no-show statistics. When done right, nobody should have to be offloaded. Even if that is not the case, AA will offer incentives to those willing to give up their seats voluntarily. Either way, your seat is safe.

American Airlines has assigned seating. Some of the seats may be blocked from assignment. There will be no-shows and missed connections so you will probably get on the flight. Go early.

No you should not be concerned, some airlines don't offer assigned seating which allows passengers to sit hopefully wherever they want to. Had it been assigned seating, when you booked your flight you would have been asked where you wanted to sit. I recommend though that when you do check in on Tuesday for your flight (or check in the night before online and print boarding pass if they allow it that early), check in as early as you can and that way when they start calling passengers by sections (A, B, C... 1, 2, 3... whatever, it'll tell you on your boarding pass) you can board in the 1st set of passengers so that way you'll have more of a selection of seats to pick from :-)