The song seems to generally be called One Eyed Jacks by Dylan fans. One of the earliest references comes from Robert Shelton's No Direction Home (1984), p. 54 [link]

Only three days after Dylan had seen him, Buddy Holly was dead. Bob and his friends studied details of the tragedy. At one A.M. on February 3, 1959, a Beechcraft Bonanza, chartered in Mason City, Iowa, took off in light snow for Fargo, North Dakota. Trouble developed in minutes, probably because the twenty-one-year-old pilot couldn't cope with the weather and instruments. The left wing hit the ground first. Killed instantly were the pilot and three musicians, Holly, twenty-two; Ritchie Valens, seventeen, a Mexican-American whose biggest hit was "La Bamba"; and J. P. Richardson, twenty-four, who was called the Big Bopper. Bob's friends and family senses a pervasive charge in him. He seemed in a tremendous hurry; all the car and motorcycle accidents indicated that his time was limited. "I was burned with death all around me," Dylan said in 1965. Gretel Whitaker, a friend from the University of Minnesota, said: "We never really expected Bobby to live past twenty-one." By the time Bob was nineteen, he had written a mournful blues, passed on to me by Minneapolis friends:

The queen of his diamonds
And the jack his knave
Won't you dig my grave
With a silver spade?
And forget my name.

I'm twenty years old.
That's twenty years gone.
Can't you see me crying,
Can't you see me dying,
I'll never reach twenty-one [12]

[12] - One Eyed Jacks by Bob Dylan. Unpublished, reprinted with permission of author.

Heylin Clinton's Revolution In the Air (2009), pp. 29-30 [link], while discussing Dylan's earliest attempts at songwriting, notes that there is not a known recording in circulation, but also adds:

One song, called either "One-Eyed Jacks," according to John Bucklen, or "Twenty-One Years," could be on the complete "St. Paul Tape," in possession of one Karen Moynihan. But it seems more like the traditional "Twenty Years Old," which Dylan recorded thirty-three years later for World Gone Wrong. Until a dub of Moynihan's May 1960 home-made tape is accessed (see my Recording Sessions 1960-1994 [pp. 1, ), we are reliant solely on John Bucklen's memory for a description of this formative effort:

"He had a list of about 100 songs that he had written, and some of them were really great. I remember one that he did... It goes 'I'm twenty years old, there's twenty years gone, don't you see my cryin', don't you see my dyin', I'll never reach 21.' Another verse is: 'The Queen of his Diamonds and the Jack of his Knave, won't you dig my grave with a silver spade, and forget my name.' It was one of those tragic things that was appropriate for the time -- a backwoods blues folk song."

Another website (The FM Club) notes that the song was "recorded at Karen Wallace's apartment, St. Paul, MN, May 1960. In Ian Woodward's transcripts of the Robert Shelton interview with John Bucklen in "Isis" [Bob Dylan News magazine issue no.] 101 [February March 2002], two of the verses are given" identical to the above.

According to most fan websites (The Official Bob Dylan Fan Site and Setlist.fm, among others), Dylan is only known to have played the song publicly twice: at Wallace's house, and once in May/June 1960 at the St. Paul, Minnesota venue The Purple Onion. According to one site, there was no known circulating recording of the Purple Onion show since at least 2001 (link) and into 2021 (link); where the list ultimately originates from remains unclear.

The other recording, Wallace/Moynihan's, is only publicly available as the "armpit tape" (linked by the original poster above -- reportedly, Brian Stibal posed as an interested seller and recorded the material with a mic tucked in his armpit). It seems as if there is a full recording on a bootleg, titled "I Was So Much Older Then Vol. 2," though I've been unable to find a copy for sale or a digital recording.

The name, One Eyed Jacks, is extraneous from the verses above, likely indicative of there being more to the song. Notably, the lyrics he sang in the tape are a bit different than those published in either book. The recording sounds more to my ear like:

I'm twenty years old
That's twenty years gone
I can't go on
I know that's awfully long
But I can't go on

I'm twenty years old
That's twenty years gone
I'll (-- skip? / pop --) never reach twenty-one
Mother, mother, I'm coming on home
Can't you hear me c[ry?] --