Yeah, my thoughts when I read their first mention of an apology was "oh, so they're going to apologize to her? I don't like them but I guess that makes sense" and then it's revealed that THEY'RE the ones who expect an apology... for what? Letting them shoot themselves in the foot???
It's more that the Prince offered to lighten the punishment he was imposing on her (the exile) if she apologized to his new fiancé for bullying her. As she refused the exile stuck much to the shock and surprise of the prince. He still probably believes Sharon was bullying the new fiancé and it was just her pride that kept her from apologizing and avoiding exile. And he also probably believes that if he is able to bring her back to the kingdom to her parents out from the exile he'll be able to win them back over as allies even if he doesn't marry their daughter, thus strengthening his claim to the throne.
Of course, he doesn't realize that Sharon doesn't mind the exile or that she is thriving as an adventurer. He probably thinks she is distraught at being separated from the nobility, her family and all the creature comforts that had been afforded to her. So his thought process is mostly likely that Sharon will beg him to rescind the exile when he tracks her down and he'll be able to convince her to publicly apologize and swear to support him in exchange.
We'll probably learn later that Sharon wasn't actually exiled because the crown prince likely doesn't actually have the power to do that or unilaterally annul the engagement and he planned to convince the King to support him on that but Sharon's mother being a good friend of the Queen would likely stop the exile while still allowing the annulment to happen as her daughter seemed pleased by it.
In short, the prince is stupid and getting played by basically everyone.
