OnSong Pro is purchased on your Apple ID. If you change devices, it may appear that you have to pay for the app again. Typically this is because the App Store has not authenticated your Apple ID and does not know that you had originally purchased the app. Tapping on the buy button will prompt you to sign in and then offer you the download for free. You could also download a free app first, sign into your Apple ID and then OnSong Pro will appear as a download link. If you had requested a refund for OnSong Pro, you may not be eligible to download OnSong Pro without repurchase due to a change in Apple's App Store policies.
If you've purchased add-ons in OnSong Pro like OnSong Console, Drawing, or other features, you may need to restore those purchases on the new device. To do this, go into the Utilities Menu by tapping on the gear icon in the Menubar. Next, tap on the Account option and choose the Add-ons option. Tap on the Restore Past Purchases button at the bottom to have OnSong Pro request purchases from your Apple ID and activate those features in OnSong Pro.
Keep in mind that if you are currently using a different Apple ID now then what you used to originally purchase OnSong Pro, since your Apple ID and purchase history are maintained by Apple, you will need to contact them directly for assistance. We have no way to access your Apple ID account information or make those changes. You can contact App Store support via this link: https://getsupport.apple.com/