Is breaking off the relationship with the young man considered a correct action to avoid the forbidden, and what is the optimal course of action in this situation, knowing that the young man is clinging to her?

You have done well by staying away from what is unlawful and its means, for Allah rewards patience and chastity. Piety is the key to relief and the facilitation of affairs. You should perform Istikhara concerning marriage to this young man. If you find him acceptable, you may send a message to him through one of his or your mahrams, while trying to lighten the burden of the dowry and financial costs on him. Seek Allah's help.