Q: When a player catches their own air ball, am I interpreting it correctly that they can only legally pass or shoot again (not dribble) regardless of whether or not the original shot was off the dribble or off the catch?
A: I am assuming that your are referring to a shot that misses the ring. This relates to team control:
14.1.3. Team control ends when: The ball has left the player's hand(s) on a shot for a field goal or for a free throw.
In this case, provided the officials deem the miss as a legitimate shot attempt, the shooter may retrieve the ball and dribble, pass or shoot since the shot ends both player and team control.