Is the questioner considered an adulterer for his act with a girl he intended to marry, and is he entitled to get her back after she became engaged to another man?

You must repent to Allah, the Almighty, from fornication. You should not expose yourself by revealing what happened between you and her. Imposing the hadd (punishment) is not a condition for repentance. Rather, sincere regret for the sin, seeking forgiveness for it, and firmly resolving not to return to it are sufficient. As for the woman, you have no claim over her. You should leave her and distance yourself from her, and seek another woman who possesses religiosity and good character.