Is the phrase, "Marry me to your daughter," essential for the completion of the marriage contract, and how can one dispel the whisperings of not having fully comprehended what was said during the contract, while being certain of having comprehended a part of it?

The formula of offer and acceptance is one of the pillars of a contract. The marriage is valid as long as the formula has been pronounced. You must disregard the whispers (waswas) about the invalidity of the marriage, because their matter is serious and may corrupt one's religion and worldly life. The greatest cure for whispers is to disregard them, pay no attention to them, and act contrary to what they call for.