Yes, the dragon got away and they had to hunt him down. Being able to stop the dragon's escape/flight takes a few particular abilities (given the speed differential and Legendary Saves).
The dragon had no intention of eating the PC (the party did not know this, also, the cleric had access to raise dead) but rather using the hostage as leverage to deal with the party. (That's how I outlined the green dragon's motivations ...). Something like an extortion deal.
One of the neat aspects of the party's tracking the dragon down was how the Barbarian (totem path) used Commune with Nature. A ritual at level 10.
The party had to focus fire on the Warlock (Archfey, Volo's) during the battle (he was riding on the dragon's back).
With the loss of that ally, the dragon needed its lair to help defend itself versus the party.
(They were able to evade or handle most of the minions...good tactics and good use of Suggestion by the bard...).