Am I obligated to continue my engagement to someone who is still far from religion, whose behavior is bad, and whom I suspect of betrayal, especially since I have concealed his past from my family, and I fear the impact of that on my future and my children?

If the young man informed others about the sins he committed, he has done wrong, because what is incumbent upon him is repentance and to conceal himself, as the Prophet, peace and blessings be upon him, said: "All of my Ummah are pardoned except those who publicize their sins." The young man’s open declaration of his sins is an act of ingratitude for Allah’s grace of concealment and an encouragement for others to commit sins.

The Sharia guides towards marrying a person of good religious commitment and character. So, observe his current state: if he repents and becomes upright, that is good. Otherwise, dissolve the engagement, for dissolving it is easier than proceeding with the marriage and then having disputes and divorce.

Do not inform your family about his sins, but if you are asked about the reason for dissolving the engagement, tell them that he is not suitable or that he is not pleasing in his religious commitment, hinting at it rather than stating it explicitly.