HDAS has the same first citation, marking it as origin unknown.
My best guess would be that it is a euphemistic variation on fucked up, but not an acronym.
Fugazi is an Italian surname that has lent it’s name to various things in the States, many eponyms for a prominent member of the Italian-American community in the late nineteenth century, including the nationwide Fugazi Travel Agency and the Club Fugazi theater in San Francisco. The existence of the name may have influenced the coinage of the slang word, although I doubt the slang is directly related to anyone of that name.
I just looked it up in Green’s Dictionary of Slang. It plumps for the acronymic origin: “Fucked Up, Got Ambushed, Zipped In.” I’m highly skeptical. Beyond the usual questioning of acronymic origins, 1980/81 is a very late date for Vietnam slang to begin appearing in print. Google Books turns up nothing from the Vietnam era except magazine ads for the travel agency.