From the notes to the Penguin Book (1959):

"The old man's courtship is an ancient joke of which country folk never seemed to tire.  In a form similar to the one we publish, the song appeared in the Musical Miscellany (London) in 1730.  It seems to be widespread in Scotland, and Sharp found it common in the West Country.  Versions have been reported from Yorkshire (Traditional Tunes, Frank Kidson, 1891, p.92; FSJ vol.II [issue] p.273), Worcestershire (Folklore X pp.173-4) and Wiltshire (Folk Songs of the Upper Thames ed. Afred Williams 1923, p.73).  Our text is amplified from the Wiltshire version. -R.V.W./A.L.L.

This version was collected by G.E. McCleay from an unnamed singer in Coggeshall, Essex (date unknown), and was first published in the Journal of the English Folk Dance and Song Society, vol.III [issue ] p.130.

Other versions on the DT:

Grey beard newly shaven  Jeannie Roberton's version; no tune given.  This was transcribed from one of the "Folksongs of Britain" series originally released on Caedmon Records, but is only an extract; the full text, also called  A Dottered Auld Carle   is given below.

The Maid I Am  In spite of its title, this is an exact duplicate of the above.

Old Shoes on and Leggin's  With tune; no source specified for text or tune.  Presumably an American version.

In the Forum:

There Was an Old Man Came over the Lea  Version from Morpeth as published in  Ancient Poems, Ballads and Songs of the Peasantry of England,  (The Percy Society, 1846); no tune given.

@courting @age @marriage @aging

There is an entry at  The Traditional Ballad Index:

An Old Man Came Over the Moor  (Old Gum Boots and Leggings)

Other titles:

The Young Lass contra Old Man
The Carle He Came o'er the Croft
The Auld Carle
I'll Not Have Him
The Old Man from Over the Sea
His Old Grey Beard Kept Waggin'
Overshoes and Leggin's
The Old Black Booger
The Old Man

There are three American versions available, with sound files, at  The Max Hunter Folk Song Collection:

With His Long Cane Pipe a Smok'n
With His Old Gray Beard a Shining
With His Ole Gray Beard A Shining

Malcolm