So does this mean he only triggers if a ship is dealt exactly one damage card? I also read this to mean he can recover a charge with each damage card dealt. I'm thinking the later because:

My thoughts exactly. The Chewie card specifies "dealt 1 damage card" and the rules say "damage is suffered one at a time," so I think you get a charge for each damage card.

With the Resistance Falcon only having 3 shields and the bomber the same, and both are high hull ships, I could see potentially triggering Chewie a few times in a game.

Yup. Since damage is dealt one at a time, Chewie should be able to charge up for each damage card that lands on a friendly ship in range (to his max of 2 charges, of course).