What's wrong with "I lead an old Dan"? Not to mention "I ride an old paint," which is perfectly clear. Cowboys in the 19th century used a lot of expressions that have disappeared from everyday life. It could have been common at the time to refer to animals with "a" or "the" -- as in "the Fiery and the Snuffy." Besides, they tended to have an excellent sense of humor and to love plays on words. Maybe "an old paint" was descriptive, and the pack horse was named Dan, and whoever wrote the song just liked the little wordplay.
This song is puzzling in a number of ways but so far no one has given a good rationale for messing with it.
I can't figure out why "throw the houlihan" in this song would be taken to mean have a wild time? Or anything except throw a rope? I always believed it means just what it says.... the singer is going to Montana to work there as a cowboy. Maybe the last job folded, maybe the singer just got restless and wanted to move on to somewhere else.
Oh, I just remembered something -- maybe he's changing his line of work (from farming). Dick Rodgers used to sing a verse I liked that went something like this:
All my life I've worked on a farm
All I got to show is the muscle in my arm.
I'll ride an old Paint, I'll lead an old Dan,
I'm goin' to Montana to throw the houlihan.