After looking up "anacrusis" (it's good to learn things), yes, I believe it's anacrusis. I haven't tried playing it, though, just singing, so I don't know how the chords fit.

Here's a YouTube of Dixie Chicken.

It's funny, but the song never sounded like there was anything odd to it to me. I guess some people grow up hearing certain somewhat cultural attributes such as syncopation, and some don't.