Is it obligatory to fast for a full month, based on a previous oath to fast it if the transgression occurred, knowing that the transgression has indeed taken place?

This type of vow is called a vow of lājāj (obstinacy or contention), and the person who made it has the choice between fulfilling what they vowed or offering an expiation for an oath (kaffārat yamīn). So, you have the choice of fulfilling the fast of the month in the summer, or offering an expiation for an oath: which is feeding ten poor people, or clothing them, or freeing a believing slave. If you are unable to do that, then fasting three days.