“I’m Going to Leave Old Texas Now”?
No words that I know of.
More Morricone, I’d say. But again, I don’t think it’s from a movie
It’s not in a minor key, but “Bury Me Not on the Lone Prairie” is mournful, and certainly has a cowboy feel.
Nope, I know that one too.
I think this one has all the do-do’s in the right places.
Have you tried humming it into Shazam or one of those other song recognition apps? I never had any luck with those, but I think with AI assistance they’ve gotten better. That’s all I got.
Too upbeat. 
I know exactly what the OP is talking about. I don’t know the proper title but I call it a “Cowboy Mosey”
Thanks, I’ll try that a little later. Right now, I need to get some sleep. 
Is it that stock music that I hear in the WB cartoons sometime as a livestock/farm motif? Can’t think of a specific cartoon, though.
Try watching the ‘Three Amigos’ clip that @GuanoLad posted upthread-- maybe it’ll help with that 
Could you record yourself humming a bit of the melody and uploading it (youtube or vimeo or whatever)? Trying to identify a song by verbal description has its limitations.
The “On the Trail” section of the “Grand Canyon” suite?
(The relevant bit starts at 12:33, if this link doesn’t automatically cue up correctly.)
Probably not this one, but I like it a lot so here it is:
“Happy Trails?”
“Doom ba da ba, doom ba da ba….”
Globetrottin’ by Top Topham?
Sounds like the theme from Gunsmoke 
Barbara Allen?