I only know the UA areas, but there may be others.

Terminal 1 Concourse C near at the top of the escalators from the tunnel has a big blue postal mailbox which I have used many times.

There is normally a full-service post office airside between terminals 1 and 2 with clerks available during business hours (won't help you) and a mail drop, but I think it is closed right now due to construction. I'm not sure if they relocated it somewhere else temporarily or not.

Assuming you're flying AA, all of these areas are accessible without exiting security, but Terminal 1 Concourse C is a pretty long walk, but the tunnel to Concourse C is worthwhile for sightseeing if you haven't done it before.

Just be careful where you walk because it is easy to make a wrong turn and unintentionally exit security while walking airside between terminals 3 and 2 and terminals 2 and 1.