Hi Doodette,
Thank you for posting in the Apple Support Communities. It sounds like you're getting messages on both devices because you're still signed in to the old device, correct? We'd love to help get the issue resolved for you. Is the new device with the same carrier, and using the same phone number? If so, did you move your SIM card to the new device?
If the new device has all of your data transferred and is up and running as it should be, we recommend signing out of your old device following the steps we've linked below.
Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac
Please let us know if you have any additional questions. We're always happy to help.
Cheers!