I pretty much have your same isue, i have my current apple id (email@me) and saw that under mail settings in icloud.com/mail apple has already given me my new @icloud.com alias. However i cant recieve or send mail using it. I made a bit of a research and found this which was very usefull:
I think apple already gave us our equivalent of our @me mail aliases with @ilcoud, but in order to use them we need to wait for the email notification they will send us when the whole process is complete, as it says on the article. This could take a few days or maybe some weeks. I hope it doesnt take long because im planing on using the new @icloud adress for work and maybe creating a new alias for my other business.
However i noticed i can use my new @icloud alias to sign in to my apple acount. I can also use it to send mail from icloud.com/mail (browser) but if i reply back from for example my gmail it gives the mail delivery failure message.