I set up a custom domain on a SHARED mode Azure site, with a CNAME for www.lakestarlodge.com, and an A record for lakestarlodge.com. Both work - I can type either into my brows
I knew that couldn't be the default behavior! Turns out, I had set up GoDaddy with DNS & "Domain Forwarding," and the Domain Forwarding (which was not left on intentionally) kept overriding the A record. So DNS was effectively being skipped. Got it all resolved!
For all the people that might still have the same problem, in my case I was using wordpress and when I created the site I choosed the Wordpress default address: mysite.azurewebsites.com. So it wasn´t a Wordpress Issue. You cand find how to solve this on this previus answer.