This is certainly connected to their issues, although I also have not received a confirmation email specifically regarding payment from Azamara and my cruise was booked in January, paid in full during the website migration through my TA.
When I asked my TA about an invoice or proof of payment from Azamara, she sent one from the travel agency and asked me to confirm on my credit card that there were payments to Azamara (which there were). That was good enough for me but I felt even better when I received an email from Azamara telling me of a change of itinerary (proof they knew I was coming aboard). Likewise, I now see the cruise and correct OBC on my profile through their website.
So, I’d recommend checking your VISA statement for proof of payment and create the log in to Azamara website. If it’s not showing on the website send in the form requesting assistance (I did have to do that once).
I would be confident it’s fine if you have proof on your credit card, but that’s me. I cruise in September and will not try to book any shore excursions etc likely until July in hopes the mess is sorted out.
Good luck!