Actually, I am supposed to have a fixed IP. And I have CNAMES configured. I will review to determine if there is some other way I can set this up, but my issue is that I would never allow my products to be offline for hours without notifying my customers.

First, I am really sorry about the impact the incident had on your service. Specifically, for your changing IP, are you currently using the reserved IP feature. You can reserve both your external IP and your internal IP.

External IP: http://msdn.microsoft.com/en-us/library/azure/dn690120.aspx

Internal IP: http://msdn.microsoft.com/en-us/library/azure/dn630228.aspx

Also as I found out, set the TTL on your CNAME very low.

Also this is a PITA if you use the @ entry in your DNS.

Why should you have to keep the CNAME low if it always resolves to the same domain on Azure?

Only because some caching DNS resolvers are broken and keep the referral cached for the TTL of the CNAME record.