Is it possible that there are people in the City of the Messenger (peace be upon him) who do not fear God to this extent, after one of them swore in the Haram that his intention was sincere, and then the questioner discovers that he is married with children?

If the engagement was merely a recitation of Al-Fatiha, then the man is a stranger to the woman, and any contact between them was unnecessary. If it involved anything forbidden, she must offer sincere repentance. The fact that he is in Medina does not sanctify him, for Medina contains both the truthful and the liars. The conclusion is that as long as the man has not officially married her, he is a stranger to her, and he has broken his promise. She should leave him and ask Allah for someone better than him. However, if he has officially married her, then he is her husband, and she has the right to ask him for a divorce due to his breaking the promise and her suffering. If he refuses or it is difficult to contact him, she should take her case to court.