"His neck was in pain and lifting his hand to it found it horribly swollen. He knew that it had a circle of black where the rope had bruised it. His eyes felt congested; he could no longer close them. His tongue was swollen with thirst..."See in text(III)
The narrator is describing the results of a successful hanging—swollen, blackened neck, bloodshot and "congested" eyes, and a swollen tongue.
"agonies seemed to shoot from his neck downward through every fiber of his body and limbs..."See in text(III)
This is an element of foreshadowing. He is feeling exactly what a hanged man would feel when his neck is broken—sudden, intense pain along the nerves extending from the neck.
"As he is about to clasp her he feels a stunning blow upon the back of the neck; a blinding white light blazes all about him with a sound like the shock of a cannon--then all is darkness and silence..."See in text(III)
At this moment, Peyton's neck is broken as he is hanged. Part III of the story consists of the thoughts and feelings that flash through his mind from the time the sergeant "stepped aside" until the moment of Peyton's death. Upon a second reading, there are numerous clues in part III that suggest this part of the story is not a depiction of reality.