Bellamy's music is not at all hard to find; just check Camsco or Amazon. Many of his Kipling settings are available on the CD double reissues "Fair Annie" (paired with "Peter Bellamy") and "Mr Bellamy, Mr Kipling & the Tradition" ("Keep on Kipling" and "Songs an' Rummy Conjurin' Tricks"); also on the excellent 3-CD compilation "Wake the Vaulted Echoes".

In addition to Badger Clark, other cowboy poets whose poems have been set to music include N. Howard "Jack" Thorp, Curley Fletcher, Gail Gardner, Omar Barker, Bruce Kiskaddon, Herbert Henry Knibbs, Arthur Chapman, Frank Maynard, Larry Chittenden, E.A. Brininstool, D.J. O'Malley, Captain Jack Crawford and Rhoda Sivell (I guess she'd be a cowgirl poet)--you can find many threads dealing with these poem adaptations here; also check CowboyPoetry.com. Someone recently recorded an entire album of Sivell adaptations she'd worked up. One-off cowboy poems that have been set include Brewster Higley's "The Western Home" ("Home on the Range"), Alan McCandless's "The Cowboy's Soliloquy", Frank Desprez's "Lasca" and Harold Hersey's "The Lavender Cowboy". "O Bury Me Not" came from the poem "The Ocean Burial", by Edwin H. Chapin. Don Edwards has set numerous cowboy poems, by these authors and others. Lots of cowboy songs originally came from poems which were published in regional papers or in popular national papers like "Youth's Companion".

Robert Service is another poet favored by songifiers.

I'd hazard a guess that a large number of broadside "songs" were actually just poems without any specific tune held in mind by the author. Settings of broadside poems by revivalist performers are legion.

Shameless plug: I've put tunes to quite a few poems, including Bret Harte's "Coyote" (with coyodel) and Badger Clark "Batchin'", "The Old Cow Man" and "Ridin'". I also did my own setting of "Unhappy Bella", which someone mentioned above.