If you delete an app from your old phone it will not be removed from your new phone.
What messages are you still getting on your old phone? It can't be text messages because its just not possible with out a working SIM. Maybe you are getting emails or messages from third party apps like skype because your phone is connected to WiFi.
If you do not have any data that you need to save from your old phone then you can do a factory reset and that will clear the phone and you should not be getting anymore messages on it.