Contacts not Connecting to Texts or Calls

I recently got the iPhone 6, and set up a new Apple ID.

I was able to upload my contacts to my phone by connecting to my old Apple ID, and then logging back into my new Apple ID.

Now, when I text someone new, the text is associated with the number, not the contact. (I am texting 555-555-5555 instead of FirstName LastName)

I also get calls from numbers instead of my contacts.

Does anyone know how to fix this?