I decided to look into this more closely, so a couple months ago, I figured out the organ solo:
As always, there's the disclaimer that I might have something wrong.
The opening organ part in "A Whiter Shade of Pale" proved more difficult to figure out, and I have only a fragment of it:
I think I might have this an octave lower than it's played, but it helps to compare it to the organ solo in "Don't Try to Explain."
There's a bit of a melodic similarity in measures three and four, but mostly the resemblance is one of rhythm. There are sections of notes with long values (often tied to the next measure) interspersed with notes of shorter values (quarter notes and eighth notes).
In Claes Johansen's book The Zombies: Hung up on a Dream, "Neil MacArthur" (also known as Colin Blunstone) even comments on "A Whiter Shade of Pale" (on page 163):
"I went to a party with Rod when that first came out," Colin Blunstone recalled. "It went on all night this party, and we played 'A Whiter Shade of Pale' all night! It was only a little record player, so we put it on 'repeat'. I just absolutely loved it. It may have had an influence on Odessey and Oracle."According to the Into the Afterlife liner notes, "Don't Try to Explain" was written by Billy Vera. The liner notes also explain that "Don't Try to Explain" was arranged by Phil Dennys and recorded by session musicians (the organist was Harry Stoneham). So while it's evident that Blunstone was familiar with "A Whiter Shade of Pale," it doesn't seem that he's responsible for the similarity in the organ parts.