Is the saying: "Whoever abandons something unlawful will attain it lawfully" correct and was it narrated on the authority of the Messenger of Allah, peace and blessings be upon him, in the context of abandoning an unlawful romantic relationship with the hope of marrying that person lawfully?

You did well to end the relationship out of fear of Allah, for love relationships between young men and women are not sanctioned by Islamic law. As for the saying, "Whoever leaves something, Allah will compensate him with something better," it was not narrated with this exact wording in the Sunnah. Rather, the meaning that was narrated is, "Whoever leaves something for the sake of Allah عز وجل, Allah will surely replace it with something better for him." And it is not necessary that the compensation be of the same type as what was left; it could be a righteous wife or even the same girl in a permissible marriage. The compensation is ultimately up to Allah.