Did you realize the melody for the Sacred Harp is the tenor line?

The only real differences I could detect by listening to the midi at your link is that Crambambuli repeats the A part & there's a somewhat different cadence at the very end. Crambambuli has re-ti-do while Sacred Harp has re-do-ti-do and extra syllable.

Here's the Southern Harmony version (melody on middle staff)

O COME COME AWAY
(from Southern Harmony)

1. O come, come away! the Sabbath morn is passing;
   Let's hasten to the Sabbath school; O come, come away!
   The Sabbath bells are ringing clear,
   Their joyous peals salute my ear,
   I love their voice to hear; O come, come away!

2. My comrades invite to join their happy number,
   And gladly will I meet them there; O come, come away!
   How there we meet to sing and pray,
   To read God's word on his glad day,
   With joy let's haste away, O come, come away!

3. While others may seek for vain and foolish pleasures,
   The Sabbath school shall be my choice; O come, come away!
   How dear the plaintive strain,
   From youthful voices rise amain,
   With sweetest tones again! O come, come away!

4. 'Tis there I may learn the ways of heavenly wisdom,
   To guide my feeble steps on high; O come, come away!
   The flowery paths of peace to tread,
   Where rays of heavenly bliss are shed,
   My wandering steps to lead: O come, come away!

5. I there hear the voice in heavenly accents speaking,
   "Let little children come to me; O come, come away!
   Forbid them not their hearts to give,
   Let them on me in youth believe,
   And I will them receive:" O come, come away!

6. With joy I accept the gracious invitation;
   My heart exults with rapturous hope, O come, come away!
   My deathless spirit, when I die,
   Shall, on the wings of angels, fly
   To mansions in the sky: O come, come away!

---

https://www.youtube.com/watch?v=X0OAYdRGDrw