"La Cucaracha" is a very old Spanish folk song, perhaps introduced to Mexico in 1818.
See thread 5841" La Cucarocha
(Name mis-spelled at the head of that thread).

It seems to have been sung in the revolution when Mexico gained independence (1821), and never lost popularity.

What verses are original? They have been revised and new ones written for a long time; take your pick.

pdq- "...thugs and crooks into power..." Not strictly true; the leaders were serious men who tried to create a popular state, but the feudal hierarchy and opportunists soon retook center stage.

A good book of corridos, some revolutionary and from many periods of Mexico's convoluted history, is Antonio Avitia Hernandez, 2000, "Corridos de la capital," Conoculta (All in Spanish).

"La Cucaracha" is not considered to be a corrido, but an old folk song. I have received a few books that might provide older verses; I will check through them and see what i can find.