I had a search in the archives of the old-time newsgroup and found this transcription of the Carter Bros & Son recording of 'Saddle Up the Gray' by Lyle Lofgren, a knowledgeable contributor to that group. I reckon it is about as good as anyone will get - and it confirms my hearing that 'riding' goes with the 'sheep' (no sexual pun intended).

SADDLE UP THE GRAY

Saddle up the grey,
And you better get away,
You ain't a-gonna get no dinner here today
Roll around the jug and put out the light,
There ain't a-gonna be no dinner here tonight.

(One verse of yaddle-daddle. I don't think it matters if you get every daddle correct here).

(Some more yaddle-daddles, with the only coherent portion being "..ain't a-gonna be no...")
Roll around the jug and put out the light,
There ain't a-gonna be no supper here tonight
Saddle up the grey, and you better get away
There ain't a-gonna be no dinner here today
Leadin' of a goat and a-ridin' of a sheep
I ain't a-gonna get back till the middle of the week
Roll around the jug and put out the light,
Ain't gonna get no supper here tonight.

Source: transcription by Lyle Lofgren. Posted to rec.music.country.old-time newsgroup.

--Stewie.