The non-Azamara tour companies all seem to "want" at least three days notice to process cruise visa paperwork, though I suspect that they could expedite for a "small fee". Keeping in mind that we'll be there from SAT through MON, the likelihood of the government folks being available to process last-minute paperwork is also low.
Regarding whether you really have to take Azamara tours only, or if the 80+ different "authorized" tour companies can also process cruise visas for you, I'd venture to say that if the "authorized" tour company cruise visa thing truly didn't work, there would have been a major outing about it on the Interwebs. As it is, multiple companies have made a living offering these "cruise visa" tours to folks on boats for years; that suggests legitimacy. As I've mentioned on this board before, I'm taking a custom tour with one of these "authorized" companies, and have grilled my tour organizer repeatedly about this topic (and my tour is NOT pre-paid- they want payment on the LAST day of the trip). If for some reason they really don't let me off the boat even with my "tour tickets" etc, then I'll just have to suffer through three days of decadence onboard :)