Several "Pretty Polly" Kentucky versions begin "In London city I used to dwell" and end "Went raving distracted and died the same night" but do not "Ship's carpenter" or her ghost.

Several versions though are complete US versions - Steve - you mentioned Pettit and I assume you are referring to Katherine Pettit, who published the first Hindman collected ballads in the Journal of American Folklore in 1907 (via Kirttedge). Here's the A version which does mention her ghost in an ending similar to but different than the "Polly's Love" broadside of c. 1820:

A. Pretty Polly; Hindman, KY pre1907; collected by Katherine Pettit

1. "O where is Pretty Polly?"
"O yonder she stands,
Gold rings upon her fingers,
Her lily-white hands."

2. "O Polly, O Polly,
Polly," said he,
"Let's take a little walk
Before married we be."

3. "O William, O William,
I don't want to go;
Your people is all against it,
And this you will know."

4. He led her over high hills,
And hollows so steep,
At length pretty Polly
Began to weep.

5. "O William, O William,
O William," says she,
" I fear your intention
Is for to murder me."

6. "Polly, O Polly,
You have guessed about right;
I was digging your grave
The best part of last night."

7. They went on a little farther,
And she began to shy;
She saw her grave dug
And the spade a-sitting by.

8. She threw her arms around his neck,
Saying, "I am in no fear;
How can you kill a poor girl
That loves you so dear?"

9. "O Polly, O Polly,
We have no time to stand."
He drew his revolver
All out in his hand.

10. He shot her through the heart,
Which caused the blood to flow,
And into her grave
Her fair body he did throw.

11. He threw her in the grave;
Straightways he did run,
Left no one to weep
But them small birds to mourn.

12. The ship sitting ready
All on the sea-side,
He swore by his Maker
He'd sail the other side.

13. All on whilst he was sailing,
The ship she sprang a leak,
And away to the bottom
Sweet William he sank.

14. There he met with pretty Polly,
All in the gores of blood,
In her lily-white arms
An infant of mine.

15. Such screaming and hallowing,
It all passed away;
A debt to the devil,
He surely had to pay.