Is it a sin to accept a new suitor, knowing that he is better than the first, after having given initial approval to the first, especially when there was no full conviction regarding the first suitor?

It is not permissible to propose to a woman who has already received a proposal from another man, unless the first suitor withdraws or gives permission. This is based on the Prophet's (peace be upon him) saying: "A man must not propose to a woman already proposed to by his brother, unless the first suitor withdraws or gives him permission." However, if more than one man proposes to a woman without the second suitor knowing about the first proposal, she is free to choose the best among them. This is similar to what happened with Fatimah bint Qays (may Allah be pleased with her), when both Mu'awiyah and Abu Jahm proposed to her, and the Prophet (peace be upon him) advised her to marry Usamah ibn Zayd. Therefore, there is no harm in accepting the second suitor and rejecting the first, and the choice should be made according to Islamic guidelines.