Is it permissible to change one's family name or add a last name, and attribute oneself to another family, knowing that everyone is aware of my situation, and the reason for this is to avoid embarrassment from people's questions?

It is not permissible for a person to claim affiliation with a family to which he does not truly belong. Claiming a lineage other than one's original, true lineage is forbidden in Islam, and the one who does so is threatened with a severe warning. The Prophet, peace be upon him, said: "Whoever claims to be the son of someone other than his father, while he knows that he is not his father, then Paradise is forbidden for him." And he said: "Whoever claims kinship to a people among whom he has no lineage, let him take his seat in the Hellfire." This prohibition is due to the ingratitude for blessings, the waste of rights, and the severance of kinship ties involved in such an act.

However, if one's lineage is unknown, it is permissible to affiliate with a large tribe or family to which one is most likely to belong, or to affiliate with the tribe of those who raised him, as an ally to them. This is based on what Hatib said to the Prophet, peace be upon him: "O Messenger of Allah, I was a man from Quraish, but I was not of their inner circle (i.e., not of their direct lineage)."