American Airlines has round-trip flights from Austin (AUS) to Rio de Janeiro (GIG) for $497-$535 with a connection. Flights November to January. Departures on Monday, Tuesday, Thursday, Saturday, and Sunday. Very few extended weekends available. No Thanksgiving dates.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.