Summary: In short, you can't have the record you want, and your DNS host is doing things The Right Way.
Explanation: It is a violation of the DNS standadards to have a CNAME (alias record / forward record) at zone apex (the empty name at the front of the zone).
The reason for this is a CNAME record cannot have the name portion conflict with any record except a DNSSec record. In a typical zone, a CNAME record at zone apex would collide with at least the SOA and NS records (and likely several others). While some DNS servers will allow this, it is a Bad Thing, and can cause hard to diagnose failures (not to mention will not work if you move hosting of the zone to a standards-compliant DNS server, such as anything BIND-based).
Either have A records at zone apex (they can be a simple web server that just throws an HTTP 302 to www). If you can get static IP numbers for your Azure server instances, put an A record for each at the apex of your zone, and create a single CNAME record called "www" that points to the apex record.
As an example :
@ IN SOA ns1.example.com. firstname.lastname@example.org. (
3600 ; )
@ IN NS ns1.example.com.
@ IN NS ns2.example.com.
@ IN A 126.96.36.199
@ IN A 188.8.131.52
@ IN A 184.108.40.206
ns1 IN A 220.127.116.11
ns2 IN A 18.104.22.168
www IN CNAME example.com.