You'll find the Leadbelly Titanic here (click). It's a very different song - refrain is "Fare thee, Titanic, fare thee well." There are some great lines in the song. My favorite verse is:

Titanic was comin' 'round the curve
When it ran into that great big iceberg
Cryin' fare thee, Titanic, fare thee well

Complete lyrics are in the Digital Tradition, version 7 of the song.

For years, I thought of "When that great ship went down" as a humorous song. My parents sang it at parties where the neighbors (?) drank a bit too much, and later I sang it at Boy Scout camp. Recently, I've heard some more serious renditions of the song, but I still think the lyrics are meant to have an element of humor to them, even when sung by the sober.

-Joe-