"Randolph the Bow-Legged Cowboy" is in Josepha Sherman & T.K.E. Weisskopf's Greasy Grimy Gopher Guts: The Subversive Folklore of Childhood (Little Rock: August House, 1995, pp. 146-47):

Version one:

Randolph the bow-legged cowboy
Had a very shiney gun.
And if you ever saw it,
You would surely make you run.
All of the other cowboys
Used to laugh and call him names.
They never let poor Randolph
Join in any cowboy games.
Then one foggy Christmas Eve
The sherriff came to say,
"Randolph with your gun so bright,
Won't you shoot my wife tonight?"
Then how the cowboys loved him
And they shouted out with glee,
"Randolph the bow-legged cowboy,
You'll go to the penitentiary!"
(Author Sherman, New York City, late 1960s-early 1970s.)

Version two:

Then all the other cowboys,
Laughed and shouted out with glee,
"Mav'rick the Lonely Cowboy,
We'll hang you from the highest tree."
(Jim, St. Patrick'sParochial School, Bedford, New York, ca. 1960s.)

"Rudolf the red-nosed cowboy" is in Mary and Herbert Knapp's One Potato, Two Potato: The Folklore of American Children (New York: Norton, 1976, p. 169).