Then why don't we also drink water as the "water of life" that Jesus said he was?
John 4:9-14
Then the woman of Samaria said to Him, How is it that You, being a Jew, ask a drink from me, a Samaritan woman? For Jews have no dealings with Samaritans. Jesus answered and said to her, If you knew the gift of God, and who it is who says to you, Give Me a drink, you would have asked Him, and He would have given you living water. The woman said to Him, Sir, You have nothing to draw with, and the well is deep. Where then do You get that living water? Are You greater than our father Jacob, who gave us the well, and drank from it himself, as well as his sons and his livestock? Jesus answered and said to her, Whoever drinks of this water will thirst again, but whoever drinks of the water that I shall give him will never thirst. But the water that I shall give him will become in him a fountain of water springing up into everlasting life.
The reference is the water of baptism. The difference between the prophecy of John 4 regarding the Baptism and John 6 regarding the Eucharist is that the liturgical part of Baptism has already been known thanks to St. John the Baptist. Fittingly, Jesus does not tell the apostles to give people water in that passage or anywhere else, and He does tell them to baptize (Matthew 28:19). He also tells the Apostles to give people the Eucharist (Luke 22:19)