The MM says that "powerful creatures can sometimes gain a white dragon's obedience through a demonstration of physical or magical might ... a white dragon defeated by a frost giant often becomes its servant, accepting the mastery of a superior creature in exchange for asserting its own domination over the other creatures that serve or oppose the giant."
It wouldn't be too hard to tweak that so that the dragon submits to whoever struck the final blow to it or dealt the majority of damage. If only a single party member speak draconic then that is another possibility. It might even be fascinated by one of the character's magic and follow them instead. The dragon might willingly go along with that character, but expect to rule over the other party members and demand a portion of the treasure.
Since it's a young dragon I would have it behave like a teenager. It'll sneak out at night to have 'fun'. Think of what a freakishly strong, super evil, and arrogant but forcefully humbled teenager would do. In the morning the party could come across a slaughtered and half eaten herd of cattle (possibly with the dragon still sleeping in the middle of the corpses with a stuffed belly), a villager who was forced by the dragon to do embarrassing things for the dragon's amusement, a merchant who was robbed of everything shiny he had (the dragon hid it under a loose scale). And, of course, it will frequently challenge the authority of whoever it decided was strongest. I expect that it would steal anything valuable they leave lying around and adamantly lie about having taken it even when it's obvious to everyone. While flying overhead (if allowed to) it might notice structures or ruins in the distance and pester its master to go visit them.
Oh, and since it's a growing dragon it will probably want a lot of food. The PHB gives prices of some farm animals:
2cp Chicken (a tasty snack used as a reward for good behaviour?)
1gp Goat (perhaps they are hard to find, or the dragon dislikes the flavour)
2gp Sheep (the dragon could complain about wool getting caught in its teeth)
3gp Pig
10gp Cow
15gp Ox
A lion, another Large sized creature (though probably smaller than the dragon) eats anywhere from 10-25 lbs of food per day, sometimes in large portions once every few days. I think that the dragon could get by on a minimum of one medium creature per week or a large creature per month. It would want double that to really feel full. The challenge would likely be finding a farmer willing to feed his animals to a dragon.
What I'm saying is to make it an NPC, not a series of ability checks or a special spell that costs money. Perhaps by the time they reach their destination your party will decide that they want to keep the dragon instead of handing it off to whoever hired them.