A “waddy” or “waddie” is a cowhand or “cowpuncher.” The origin of the term is unknown.
 
 
Google Books
Cowboy Lingo
by Ramon F. Adams
New York: Houghton Mifflin
2000
1936 (original copyright)
Pg. 22:
The cowboy was known, too, by such slang names as “ranahan” (which really referred to a top hand), “saddle-warmer,” “saddle-slicer,” “saddle-stiff,” “leather-pounder,” “cow-poke,” “cow-prod,” or “waddie,” but the most common term used in the cattle country was the simple title of “cow-hand” or “hand.”
Pg. 154:
An early name for the genuine rustler, one faithful to his illegal art, was a “waddy”; later this term was also applied to any cowpuncher.     
   
(Oxford English Dictionary)
waddy
U.S. slang.
Also waddie. [Origin uncertain.] 
A cattle rustler; a cowboy, esp. a temporary cowhand.
1897 E. HOUGH Story of Cowboy 279 A genuine rustler was called a ‘waddy’, a name difficult to trace to its origin. 1927 J. LOMAX Cowboy Songs 374 He rides a fancy horse, he’s a favorite man, Can get more credit than a common waddie can. 1931 W. ROGERS in S. K. Gragert Will Rogers’ Weekly Articles (1982) V. 470 You town waddies know what a Combine is?
       
(Historical Dictionary of American Slang)
1893 Griggs Lyrics 15: And [sic] waddy objecting, was shot in the breast. Ibid. 254: In speaking to his comrade, [the cowboy] calls him waddy; when talking of one, he refers to him as puncher. 
1908 William M. Raine Wyoming 231: He was a noted “waddy,” or cattle rustler.
1922 Philip A. Rollins The Cowboy 313 [ref. to 1892]: The movement also had militant apostles in the “waddies,” men faithful to the illegal art of rustling.
 
5 August 1926, Iowa City (Iowa) Press-Citizen, “Lurid Cowboy Fiction False,” pg. 14, col. 1:
Lurid fiction tales of chaparajosed cow “waddies” galloping recklessly across the prairies, or shooting the buttons from some easterner’s spat at 70 paces with notched “.45’s” have drawn a protest from Charles D. Frost, a rancher of Bozeman, Mont.
 
24 September 1927, Daily Northwestern (Oshkosh, WI), pg. 2, col. 3:
DIALOGUE OF COWBOYS
GREEK TO STRANGERS
 
San Angelo, Tex.—(AP)—A dictionary would be about as worthless as a song in a hurricane to a New Yorker trying to find his way around the ranch country of the west.
 
Cowboyese, the dialect of the ranges, is as intricate and snappy as New Yorkese and changes almost as rapidly. Some of the terms used in the pioneer days have come down unchanged through the years, but other influences—mainly that of the cavalry in which most of the cowhands fought in the world war—are apparent in the dialect.
 
What would a native of New York’s East Side do if confronted with a conversation like this:
 
“The top screw mounted his cutting horse, and, followed by a group of chuck eaters, started to trail a bunch of cattle. The corral rope was on his saddle, next to the sougan, and as he placed a brain tablet in his mouth, his mount began to swallow its head and soon turned the pack.”
 
A “top screw” is a ranch hand who has been on the ranch for years and knows the business of that particular ranch from top to bottom. A “waddie” is another name for the same individual.
 
10 July 1968, New Mexican (Santa Fe, NM), “Spanish, French, Indian, English—All in Cowboy Lingo,” pg. 4:
Waddy
Another name for cowboy. Originally meant an extra hand used to fill in during round-ups, later applied to any cowboy.