Does the phrase, "in dreams' projections" mean he relives these scenes in his dreams?

It certainly can mean that. But in the context of the poem, what the narrator is doing is preparing to draw his audience deeply into the stories he is about to tell it of his days as a surgeon during the Civil War. His memories are "dreams" not in the sense of being false or imaginary, but in that they constantly haunt his mind. The terrible memories he relates to his immediate listeners are more real than the ever-changing world that quickly moves on, forgetting the horrors of war and other traumatic events:

But in silence, in dreams’ projections,
While the world of gain and appearance and mirth goes on,
So soon what is over forgotten, and waves wash the imprints off the sand,
With hinged knees returning I enter the doors, (while for you up there,
Whoever you are, follow without noise and be of strong heart.)

Notice also the contrast between the solidity of the "dreams' projections" and the transience of the everyday world of flux. These memories, these "dreams" are more real than the fleeting world around us. Indeed, in telling his story, the old man is consciously breaking off from the world of "gain and appearance and mirth" to reenter the doors of the military hospital as he relates his experiences to the assembled throng.
https://www.poetryfoundation.org/poems/53027/the-wound-dresser