All "engaged" means is that the unit cannot move for the rest of the phase; it does not stop them from doing something that engages them. So, an engaged citizen cannot Migrate, Explore, Harvest (gets moved to harvest icon), or Construct (gets moved into new building).
What you can do in a single player turn during the Action Phase is listed on the back of your player screen, and on page 5 of the rulebook:
*In the order of your choice:
- Play one action
- Use a character or progress card
- Use a market or a port
So every player round, you may use one market, or one port, not both. You can do this every round of the Action Phase, but each market and port can only be used once.