Why do I receive text messages and missed calls on my old Iphone. How do I stop this?

When someone calls me on my new phone, it shows up as "missed call from ___".It doesn't ring on my on my old phone, but after I answer the call on my new phone, it says missed call on the old phone, if that makes sense. I receive text messages from people on my old iphone as well. I use it as my iPod because it could drain the life out of my new phone, so I don't necessarily want to reset the entire phone. I have cellular data off and I'm with Verizon.

It sounds like Verizon hasn't totally deactivated your old phone. If you have one number on Verizon, only one phone is supposed to work at all on that number. The only way to get this fixed is to call them and explain the problem, or bring both phones to a Verizon store.

It has nothing to do with Cellular Data but with the iTunes/iCloud account. You are using the same phone on both accounts.

Go into the settings on your old phone and remove your phone number and then also uncheck your phone number under iCloud.

However, it could be beneficial to still be able to send/receive texts from that (I'm guessing this happens when you are connected to Wi-Fi and that the messages are from other iPhone users).