WD, you're right that the word is spelled ('correctly') several ways. I'd imagine the derivation is from the French "chanter" ("to sing") or "chantez" (imperative: "sing") or "vous chantez" ("you sing"). Why would the shanty cabin give its name to a work chant or song?
Anyway, given that I lean toward the French derivation theory, I prefer the spelling that's closest to "chanter" or "chantez." Either way, it would be pronounced "shanty," though. (The French "ch" is pronounced "sh.")