Fixing iMessage/Message to not use email address (OSX and iOS)
Your email address, which is your Apple ID, is what ties everything together. That has not changed since iMessage was introduced.

You can change where the message is coming "from" by going (on iPhone) to Settings > Messages > Send &amp; Receive > and at the bottom under "Start new conversations from", adjust the checkbox to change what shows up as the "from" to your recipients. Most people would use their phone number here.

You should ensure the same setting is selected on your Mac. Go to Messages menu > Preferences > Accounts > select the account which says "iMessage" in small type below your email address on the left side, then on the right side, click the Settings tab and at the bottom, adjust the "Start new conversations from" setting.

The new feature you are referring to is SMS Forwarding, and is mostly unrelated to the above. It allows you to set your phone to forward messages, received on the phone as SMS, to your Mac as if they were iMessages. Replies made on your Mac are sent back to the original sender as SMS, again through your phone.