A. L. Lloyd popularized this rudely amusing song (Roud 1052) in the '60s. The DT version is mostly his, collected from Bob and Ron Copper.

Lloyd in 1971 referred to a broadside from "Swindells of Manchester" printed "nearly a century and a half ago, in a slightly different version with a less stern moral."

This broadside seems not to be online.

The song's earliest appearance, however, appears to have been in

"Daniel Cooper's garland, containing four new songs. 1. Daniel Cooper. 2. The farmer and the mole-catcher. 3. G. in tears; or, All alive at L. 4. Roger and Sue."

The British Library dates the pamphlet to "?1765," which seems to be the default date for a great many of its undated garlands.

The garland was "Sold by M. Viner, and M. Nailer, in Bristol; W. Williams, in Monmouth; J. Blunt, in Ross; J. Price, in Kington; P. Hodges in Hereford; J. Bence, in Wotton-Underedge; and M. Cook, in Glocester ([Bristol])."

In other words, it was potentially widely distributed in the West of England.

All four songs in the "Cooper" garland are bawdy. Regrettably, however, the two pages of "The Farmer and the Mole-Catcher" have been sliced out (along with most all of the otherwise obscure "G. in Tears").

But a second early text, as luck would have it, appears in "The Frisky Songster, A New Edition" (“Sold by the Booksellers in Town and Country, 1776”). Presumably this printing is about the same as the missing "Cooper" text. The tale is located in (Old) Malden, in Surrey, now a London suburb.

I've separated the solid text into stanzas; the "safe in my trap" stanza is clearly misplaced, and should appear immediately after "took her behind.":

SONG LXXIX

THE MOLE CATCHER

In Malden, in Surry, at the sign of the Plough,
There lives a young couple, as I shall tell now,
And he had a wife that was buxom and gay,
And she with a farmer was used for to play.

The Man was a Mole-Catcher by his Trade.
He went to the field with his traps and his spade;
A catching of moles from morning till night,
The farmer he used for to play with his wife.

The man had a jealousy of the thing,
He hid in the bake house to see him come in,
Then seeing the farmer come over the style,
That made the Mole-catcher begin for to smile.

He came to the door, and this he did say,
Oh, where is your husband, good woman, I pray?
A catching of moles, you need not fear:
But little she thought her husband was there.

The farmer immediately pull'd her down in his lap,
Ay, says the Mole-catcher, I've you in my trap;
Then up stairs they went for to fill their design.
The Mole-catcher followed soon after behind.

Then out of the window her head she then put,
The farmer immediately pulled up her coats;
He could not come at her before, as we find,
So he was obliged to take her behind.

The Mole catcher the farmer's coat did tear,
For he was resolv'd to examine his ware:
As he was a peeping, he saw a long pin,
He said to his wife, my dear was he in.

O yes, with blushes then did she say,
O yes, he was in, but a little way;
O that is enough, he said to his wife.
This is the best mole e'er I caught in my life.

Just as they were in the midst of their sport,
The Mole catcher caught him fast hold by the coat;
Saying, you villain, what would you be at,
I think now I have got you safe in my trap.

I'll make you pay for ploughing my ground,
And as for the money, it shall be ten pound;
The farmer said, the money I don't mind,
For it has not cost me past six-pence a time.

So now to conclude with some of the loss,
The farmer's obliged to go to the Cross,
A spending his arse, I know not how,
He durst not go to the sign of the Plough.

The lyrics exploit the connotations of the word “plough.” “The [sign of the] Cross” may imply a pun on “cross-ploughing.” (Oxford’s earliest date, however, is about a decade later.) To “spend one’s arse” (evidently ‘to spend much or too much money’) isn’t otherwise recorded.

More information about revival recordings, many stemming from Lloyd's, is here:

https://mainlynorfolk.info/lloyd/songs/themolecatcher.html