As long as the phone is still functioning, and you can turn it on and access the phone, you can get the files from it. Service or no service.
One way is to access the internet, via WiFi, on the phone and upload the files to a cloud or google account from the old phone, and then download them to the new one.
Another way would be to connect the old phone to a computer, via USB cord, and move all the files from the old phone into an empty folder on the computer. Once moved, disconnect the old phone from the computer and connect the new phone via the USB connection. Then move the files from the computer over to a folder on the new phone.