If there is no screen-protective film installed and no case and you’ve cleaned the iPhone display screen, then it would seem your display has failed, and you’re getting ghost touches.
There’s no good way around that short of a repair, and you probably replaced your iPhone because you didn’t want to repair the old one.
If you have an iCloud or with Mac or Windows backup, you can restore that.
Your app purchases and subscriptions are associated with your Apple ID, and will transfer or can be re-downloaded. Whatever iCloud services and contents have been configured will also be available and preserved.
If you have local data and have no backups and have little or no iCloud data, then that local data is going to be a problem recovering and transferring. A repair of the old iPhone may or may not recover access to that data.