Green’s Dictionary of Slang

(US) to court young women; thus go a-gallin, go gallin, to go courting.

1840		T. Haliburton Clockmaker l. III 80: Few created critturs go a-gallin’ so early as he does.
1920–23		in J.M. Hunter Trail Drivers of Texas (1963) I 333: Going courting is ‘goin’ gallin’.’.
1936		R.F. Adams Cowboy Lingo 217: When a cowboy ‘went a-wooing’ he was said to be [...] ‘cuttin’ a rusty,’ or ‘gallin’.
1953		Randolph & Wilson Down in the Holler 246: gal: v.i. To seek feminine society. ‘Tom he always goes a-gallin’ of a Sunday.’.