Does divorce occur if the husband and wife agree that she will not shake hands with male relatives, then the wife shakes hands with one of them out of necessity, knowing that the husband had a fleeting thought about divorce occurring if his wife were to shake hands with a man, but he does not recall whether he actually uttered it or not?

Divorce does not occur through doubt about uttering it, because the certainty of marriage is not removed by doubt. One must strive against their self and abandon obsessive whispers. Furthermore, shaking hands with a non-mahram (definition) woman is not permissible, due to the Prophet's (peace and blessings be upon him) saying: "For one of you to be stabbed in the head with an iron needle is better for him than to touch a woman who is not lawful for him." And shyness that leads to falling into something forbidden or neglecting an obligation is blameworthy shyness.