Well, Amos , the words and tune I know to the Rocking the Cradle song, which obviously lead straight to "Git along little dogies" are:

Oh sweet baby lie easy
Your own daddy will never be known
And it's weeping and wailing
And rocking the cradle
And leave me a baby that's none of my own.

Nothing either specially English or specially Irish in that, no evidence of Irish immigration as opposed to English. What version do you have that points to an Irish singer in Texas carrying the song? What is the obviously Irish version you are talking about? Let's take a look.