On the old phone, go to Settings, tap your name, ICloud, iCloud Backup and turn it on and let it back up COMPLETELY. iPhones rarely “freeze”, it’s the user that gets impatient and expects things to happen instantly. And if you have to pay $.99 to get 50Gb of iCloud storage, just do it. Then once it completes go to the new phone, Settings, General, Reset or Transfer and then Erase All Content and Settings. Once it’s done you can either set it up using the quick setup (which has always worked for me) and be patient to let it finish, or restore from iCloud and point it to the backup you just made.

Either way you have to let the phone do it’s thing and download everything. If you don’t have fast Wi-Fi that’ll make it worse. But trust the process and all of your messages will be there when it finishes