Am I sinful for swearing and calling upon God as a witness to my love for my husband’s mother, when I am not certain of the sincerity of this love, and what is the ruling on supplicating for her long life and healing, and what must I do if there is sin in the oath?

Showing courtesy to people and expressing love to them to win them over and conciliate their hearts is a legitimate act, and it is characteristic of the believers. This is indicated by Imam Al-Bukhari's chapter heading in his Sahih (definition): "Chapter on Courtesy with People." Aisha, may Allah be pleased with her, clarified that the Prophet, peace and blessings be upon him, softened his speech to a man whom he described as "the worst man of his tribe," justifying this by saying that "the worst person in status with Allah is one whom people abandon to avoid his evil." This principle is conditioned upon not involving lies or swearing an oath upon them, for lying is forbidden, and swearing an oath upon it is a false oath (Yameen Ghamoos). If the sister feels some love towards them, even temporarily when hearing kind words, she is truthful in her oath and incurs no sin. As for how to speak with them, let it be with good words and kind speech, in compliance with Allah's saying: "And tell My servants to say that which is best. Indeed, Satan incites [discord] among them."