Both new and old IPhone are in service?

Today I got my new iPhone 6s before this I had an iPhone 5c. I transferred everything from my iPhone 5c to my desktop (not iTunes) and then turned it off. I then followed the instructions on activating the iPhone 6s, I have Verizon by the way. So after all was complete I had service on the IPhone 6s and realized I wasn't receiving any texts or phone calls. After a few hours I turned on my old IPhone 5c which i came to realize still had service and all the text/phone calls I received appeared. I'm very confused both phones can make calls and send texts. Is there steps I missed? Also on the iPhone 6s it's showing an email as the contact through text how can someone remove this? Please advise

Did you follow the directions that Verizon gave you when you were transferring service to your new phone?

I would contact Verizon. They are the only ones who can fix this problem with your phone.