Hi there - to be quite honest I was expecting your post! I did not want to confuse you first time around by going all through the following just in case you already had the number or Almoradi had a different system.
OK your problem is that you cannot register at the health centre without a social security number (this is NOT the NIE number)
You need to go to the social security offices in Calle Obispo Rocamora no 43 (check out Google maps) its near the tax office. Avoid Tuesday its market day and parking will be horrendous.
usual routine take all of your papers including copies, take your number and wait your turn.
Dont get confused and go to the other office in Orihuela (like I did on advice of my health centre) and waste 2 hrs there!
Good luck.