Yeah, and thanks for the spoiler. 
4+20 from Deja vu: Crosby, Stills, Nash & Young
sorry for caps, cut and paste:
…
MORNIN’ COMES THE SUNRISE AND I’M DRIVEN TO MY BED
I SEE THAT IT IS EMPTY AND THERE’S DEVILS IN MY HEAD
I, EMBRACE THE MANY COLORED BEAST
I GROW WEARY OF THE TORMENT, CAN THERE BE NO PEACE?
AND I FIND MYSELF JUST WISHIN’ THAT MY LIFE WOULD SIMPLY DECEASE
Also, that great spooky a capella in/from Oh Brother, Where Art Thou, where the singer ispleading with death for just a little more time. I’d look it up but iPad is cranky about reloading empty pages.
That would be Ralph Stanley. (Saw him live about a year ago, and he did that one a cappella, just like in the soundtrack… and he added “thank you” to the end.)
I don’t know the song she was singing, but Amy Winehouse’s video of the funeral was pretty prescient-along with her song about refusing to go into rehab. Its almost as if she wanted to die.
I suspect that things ended very badly for the protagonist of this song - Please Take My Scissors Away
Johnny Cash’s “Sam Hall” - funny as hell. 
And his “Give My Love to Rose” - a big enough tearjerker that it should count, even if the narrator is simply repeating what the dying man said to him.
Joe Diffie’s "Prop Me Up Beside the Jukebox When I Die" is worth a mention.
Frank Sinatra’s “It Was A Very Good Year”,even if the singer’s death isn’t quite imminent.
And that doesn’t even get into opera - there’s a ton of arias by the soon-to-die. One particular favorite is Aida and Rhadames pledging their undying love to each other, at the top of their lungs, in the tomb where they’ve been placed to slowly suffocate.
Whoops! Two more from Andrew Lloyd Webber:
Evita starts with her death and Che singing Oh What A Circus, then she rises from her coffin and joins him. It ends with the dying-or-death Evita singing her Lament.
The musical Sunset Boulevard starts with the dead Joe Gillis singing I guess It Was 5 a.m.
Personally, I prefer this version. (Frank Tovey)
Another just-about-to-be-executed song
Tony Christie - I did what I did for Maria
I Hung My Head - guy accidentally shoots someone, hangs his head, gets hanged.
Sting originally, I believe. Johnny Cash covered it.
There’s a haunting song called Little River that I first heard covered by a folksinger named Cindy Mangsen. It’s a lament by a drowned sailor who’s now (apparently) a spirit:
“Spring comes warm to Little River, storm comes black
I was headed home when the Indian Giver took me back”
And for something slightly different, there’s always James Taylor’s “The Frozen Man”.
He was dead, but they brought him back.
“I know what it means to freeze to death
To lose a little life with every breath
To say goodbye to life on earth
And come around again”
Salamander Crossing did a lovely adaptation of Tennyson’s Crossing the Bar. I’m not sure the narrator is explicitly dying or just contemplating it, though.
I think Rick Rubin found every damn song there ever was about dying cowboys for Cash to sing on his “American Recordings” albums. Hell, just get the whole set. They include, btw, "Let the Train Blow the Whistle When I Go".
Bobby Bare’s “Dropkick Me, Jesus (Through the Goalposts of Life)” (written by Shel Silverstein):
A lowly benchwarmer I’m contented to be
Until the time that You have need of me
To put on the big scoreboard that shines from on high
The Great Super Bowl way up in the sky …
whaddya know, Johnny Cash covered it.
And his Danny Boy is the best ever, too. Yes, the lyrics he used make it fit the category.
What the hell was it with Johnny Cash and dying songs.
Does The Streets of Laredocount? Technically the singer isn’t dying, but reporting the words of a dying man.
Having just come from the Heinlein thread, and recently posted the “Songs that can be sung to ‘The Ballad of Gilligan’s Island’” thread, I should long ago have thought to mention “The Green Hills of Earth,” from the very inspiring short story that gives all the necessary background.
I’d heard the title mentioned, but in the context of that mention, which was by Dave Barry IIRC, I thought it was a joke, not a real song.