Lyric discussion by jsaul 

I think some of the other interpretations are most likely spot on. That said, I wanted to suggest a slight variation that could be wrong.

I suggest that the line "So in love the preacher's face turned red" should be taken to mean that the woman was so pregnant that everyone, even the priest, could tell they were having a shotgun wedding (as in they were forced into it because she was pregnant).

I take the "Soon everybody knew the thing was dead", to refer to the baby which could have died in childbirth or a miscarriage (or SIDS but that's not really the point). Once the baby is gone, the couple has no reason to stay together so she goes to Haiti for a quickie divorce.

Burt Bacharach wrote a song called "Mexican Divorce" played by The Drifters (as well as Ry Cooder) which I imagine Fagen and Becker were familiar with, so I think they decided to put their own spin on it. I had initially thought the 2nd verse was about getting an abortion, but that doesn't seem consistent.

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