[link to original on tumblr]
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I discovered that I have to use the version on The Pet Sounds Sessions as my main referent because that version is at the original speed. The version on Pet Sounds is sped up, so it's in-between F major and F# major. At the original speed, it's in F major (which works well with what I mentioned about keys and unity here). Incidentally, They Might Be Giants' version is in F# major.
Some of the notes in the melody are Bs, which is an accidental in F major. In the first verse, they're all on verbs too:
Where did your long hair goIt's an interesting way to add emphasis.
Where is the girl I used to know
How could you lose that happy glow
Oh, Caroline, no
I figured out the instrumental part near the end, which is pretty close to the melody in the vocals, and I continued the phrases to the end, instead of fading out. I didn't really pay any attention to what instrument it's on though. At this point, I'm concerned more with accurate notes.
In one phrase of that instrumental part, the melody is sort of an inversion of the bass part. Much of the bass part is the repeating phrase G C A G F G, and that instrumental part has C G F G A (corresponding to "Oh, Caroline, no" in the lyrics). So there's something of a trill between G and F either preceded or followed by an A.