Okay... I figured this out. Kind of...
On my phone (iPhone 4s) I went to Settings > Messages > Received At
I have my phone number and my email address listed. I tried to remove the email address, but it would not let me do it. The REMOVE THIS EMAIL button would not respond.
So...
I then went to Settings > Sounds > New Mail
I selected 'None' for the alert tone. Problem fixed! No more double alert tones!
But...
This could be a problem for people who like alert tones for incoming email. It's not a problem for me, because I don't have my email pushed to me (I check it manually when it is convienent to me). I do this because I did not like my phone constantly 'binging' everytime I get an email, and helps conserve battery.
The real problem is why the REMOVE THIS EMAIL button will not respond under the Messages Settings.
Hope this will help some of yall.