I had the same problems two weeks ago. It looks as it is a problem, where you are at home. My husband, who is an IT Pro, detected the problem, that when your home country is not one of those listed in "Country Selection" on the end of Azamara website, it will not work.
We live in Austria and if we tried the booking with an adress in Germany - that worked. So we booked with an Travel Agency. And everything is fine now - I see the booking also in my account on the website.