The recording really helped, since verses start with "Ay." That identified the particular song. "Morena" is used in some versions rather than "morenita," and often used in the chorus.
Unfortunately, I found only part of versions, and although they show the variety of the verses, they don't add up to a singable unit.

The verses humorously characterize or praise a dark-complexioned girl from various towns and districts. I have found a few verses only, in the collection "Coplas del amor feliz," "Cancionero Folklorico de Mexico," Margit Frenk Alatorre, vol. 1 and II of five. This folk song seems to be widespread in southern Mexico, with many variations.

Verses are often borrowed from completely different songs and inserted as a change of pace or to insert a double meaning; I have copied a couple that are cited in Frenk Alatorre's treatise.

LA MORENITA I
(Coll. Guerrero, 1930, entry I-1491)
Ay, morenita!
morenita de los tules,
échame tus brazos, mi alma,
con esas venas azules.

(Coll. Huetamo, entry I-1006)
Ay, morenita!
morenita de Huetamo,
no tienes que preguntar:
ya sabes cómo me llamo.

(entry 1841, Michoacán)
Ay, morená!
morená, la de allá arriba!,
desde que dejé de amar,
aunque me cueste la vida,
te dejé de señalar (sic).

(entry I-1296, Guerrero)
Morena, morena,
morena del mar,
me la retellevo,
me la he de llevar.
["La morenita"]

(entry I-1858, Michoacán)
Ay, morená!,
morená la allá abajo,
no te dejaré de amar,
aunque me cueste trabajo.


(entry II-3809)
Ay, morenita!
morenita del Bajío,
todos tienen sus amores,
só a mí me falta el mío.


A different song from Veracruz
(Entry III-7407)
Ay, morenita linda
de la costa caliente!
La tierra de los jarochos
es tierra que nunca miente.

Sometimes verses are loaned from one song to another. Unfortunately, complete songs are not printed in this treatise, aimed at scholars rather than singers. One version was collected in Guerrero, two in Michoacán, with variations.
A borrowed verse that sometimes appears in "La morenita" -
(entry I-2362, Mich.)
Qué bonito sombrerito,
que rueda la cuesta arriba!
Esa prieta me la llevo,
aunque me cueste la vida.

another:
(entry I-191, Mich.)
Me gusta la leche,
me gusta el café,
pero más me gustan
los ojos de usted.
(double meaning here)

I hope someone can catch the words sung by the group on the cd. I can read a little Spanish, but my 'ear' is hopeless with Spanish.