Since the Eucharist is the Body of Christ, when we eat it, are its elements still existing as our Savior, or are they digested and become part of our own body? Do we become part of Jesus’ body? Do we become part of each other's bodies in Christ?

In John 6, the Lord Jesus teaches a kind of mutual indwelling, for He says, "Whoever eats my Flesh and drinks my Blood remains in me, and I in him” (John 6:56). Thus, while this process is mysterious and not easily reduced to mere human language, the mutual indwelling is very real, such that we are in Christ, and He is in us.
 
At one level, the Eucharist, the Body, Blood, Soul and Divinity of Christ, is food. As Jesus says, “For my flesh is true food, and my blood is true drink” (John 6:55). As human beings, the food we receive, in this case the Holy Eucharist, is wonderfully assimilated into us, and becomes the very stuff of which we are made. That is, our food becomes the very building blocks of the cells in our body. And thus the very food of Jesus’ own Body, Blood, Soul and Divinity, becomes part of our very substance. In this sense, St. Augustine says, “Christian, become what you are” (cf Sermon 272).
 
But, as all the Fathers of the Church note, unlike every other food we receive, the fruit of Christ is both living, and greater than we are. All other foods we receive, are dead. But since Christ is alive, it is not merely that we take Him into ourselves, but even more, that he takes us into Him, making us a member of His own Body. And this assimilation of us into him is far greater than our assimilation of him into us. Thus, we most properly speak of becoming a member of his body, for that is the greater effect of Holy Communion. But it is not untrue that he also becomes one with us, as the quote from John 6:58, above, teaches.
 
Ultimately, our oneness with each other is in Christ. And thus, while avoiding overly physicalist notions, we can say that we have communion in and with one another, in Christ, for we are all members of the one Body, and when we receive Christ, we receive the whole Christ, which includes all his members. This is why faith and orthodoxy is also essential for Communion. For one to be truly a member of the Body of Christ, requires that one live in union with all the members, and with its head. Ultimately the Church is Unus Christus, amans seipsum (On Christ, loving himself – St Augustine, Homilies on 1 John, 10, 3)
 
Thus, properly understood, and with necessary distinctions, the insights noted above are correct.