Yes - it is possible to get only one. As a matter of fact, I didn't realize you could get both . Starting with the iPhone, if you go into settings, messages, is it set as the phone number where you set up 'Receive At'? If not, set it to the cell number. Click on the Receive At option and see if there is also an email address added. If there is, try removing it. Check the settings on the iPad. Let me know if that helps.
Good Luck - that must be very anoying.
I did check my iPhone since I have access it to it right now and it shows the following:
You can be reached for messages at:
Phone 1 (xxx)xxx-xxxx (grayed out)
Email firstname.lastname@example.org (selectable)
Add Another Email...
If I select the gmail account it pulls up another screen with the email address and Remove This Email grayed out (unselectable)
I also have a Caller ID section which, when selected, shows both the phone number and the email. The phone number is checked. There is no option to remove anything from this screen.