When we accept Christ's offer to come to him, how does He ensure we never hunger and thirst again?
I do not know exactly how God will ensure we do not thirst again. That may vary from individual to individual. I would not presume to limit the way God can do things.
I do know that the promise is fulfilled in part in the Holy Spirit, the word of God. Since he is promising spiritual rather than physical filling, we must go to the source of spiritual satisfaction.
“Whosoever drinketh of the water that I shall give him shall never thirst; but the water that I shall give him shall be in him a well of water springing up into everlasting life.” (John 4:14)
“Be not filled with wine, wherein is excess; but be filled with the Spirit; Speaking to yourselves in psalms and hymns and spiritual songs, singing and making melody in your heart to the Lord; Giving thanks always for all things unto God and the Father in the name of our Lord Jesus Christ; Submitting yourselves one to another in the fear of God.” (Ephesians 6:18-21)
This latter verse says we are filled with the Spirit when we sing to God, give thanks, and submit ourselves to one another. Perhaps the church and personal study, then, are two ways that Christ ensures that we are filled.