Is it a condition to drink one's fill of Zamzam water for a supplication to be answered, if the quantity is limited and given as a gift, and one wishes to distribute it among family and friends for blessings?

Being filled to satisfaction (al-tadallu’) with Zamzam water is not a condition for attaining what it was drunk for. The hadith: "Zamzam water is for what it is drunk for" is absolute. Some hadith masters authenticated it, while others deemed it good (hasan). Many imams acted upon it and achieved their objectives. As for al-tadallu’, it is mentioned in another hadith: "Indeed, the sign between us and the hypocrites is that they do not drink their fill of Zamzam." This is an authentic hadith. It is sunnah to drink Zamzam water with the intention of fulfilling needs and to drink one's fill of it.