I just went through this and the fix is easy. All you need to do is login to your Web Client where you added the email address and create a filter for THAT email address. The filter will be when the "From" address is the email address that you are sending from (the one that you added), DO NOT send to handheld.
For example, if you added
I am guessing that you are getting messages from that email address to that email address every time you send an email.
Just put that whole email address in the filter and you should be good to go.