I finally found the song I was looking for; it is not one of the ones above. We lived in Santa Fe, and she sang in Spanish although that was not her native language.

Tecolotito orguloso
Tecolotito orguloso
Que haces in ese Sotillo
Que haces in ese Sotillo
Hoo?

Cu, curu cu, cu, cu,
Pobrecito animalito
Tiene hambre tecolotito,
Hoo!

Mirando los borachitos
Mirando los borachitos
Empinarse la botella
Empinarse la botella
Hoo!

Cu, curu [etc.]

In Mary van Stone and Alice Corbin, Spanish Folk Songs of New Mexico (1926). The translation in the book is impossibly strained. It more or less keeps the scansion but not the meaning of the Spanish words.
I understand the chorus "cu, curu, etc." means both a sound an owl makes and something a mother would say to a baby. And the verses are full of diminutives that are a little hard to translate:

Proud little owl,
What are you doing in the [cute little] thicket, Hoo?

I am watching the [cute little] drunks
Tip the bottle. Hoo!