Both new and old IPhone are in service? Help?!

Hello,

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 am 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

Thank you

Read more: Both new and old IPhone are in service? Help?!