"What I'm getting is that there is no basic Country or Western song structure the way there is with a 12 bar blues. Is that right?"
Right, certainly for songs (= vocals).
The great (overwhelming?) majority of country songs are 3 or 4 chorders, usually I-IV-V, I-II-V, I-II-IV-V, or I-IV-V-VIm in some order, but that's also true of rock and folk, and none of these genres has a this-is-what's-used-90%-of-the-time structure like blues or doowop.
Country music is more defined by subject matter and feel (e.g. plaintiveness in some songs) than by the music itself. It often has an artless, down-home style of expression that at its best is profound in its simplicity. The music behind it has a complementary simplicity, but not a cookie-cutter pattern.
So the trick is to explore appreciating country's strength - uncomplicated emotional expression that sometimes grabs right at your heart - and not get swallowed up by its weakness - plain-Jane musicality that can be like reading a 2nd grade textbook to those who have developed a taste for more intricate musical forms.