Is my action valid after I intended to pray the Fajr supererogatory prayer, then changed my intention to the Fajr obligatory prayer when other worshippers entered, knowing that I had not yet stood up for the prayer?

Your prayer is invalid. Your change of intention from the Sunnah of Fajr to its obligatory prayer invalidated the Fajr Sunnah, and the obligatory prayer was not valid because you did not intend it from the beginning. Therefore, you must repeat the obligatory Fajr prayer.