Sync is not bringing anything across
Sync stays off until you switch it on, so a fresh install has it off on both devices. Turn it on in Settings, then Sync, on each device separately.
Both devices need to be signed in to the same iCloud account with iCloud Drive on. Nothing you have already typed is left behind when you switch it on: it uploads what is there rather than starting you from empty, and a first sync of several months can take a few minutes.
I paid for sync and it is asking again
The Mac and the iPhone are one app with two platforms rather than two products, so $14.99 covers both. If the second device is asking for money, it is almost always signed in to a different Apple Account than the one that paid.
Check the account in System Settings or iOS Settings, then use Restore Purchases in Settings, then Sync. No receipt or email address is needed, because Apple holds the purchase, not Cadence.
Getting your data back out
Settings, then Data, then Export. That writes one JSON file with everything you have typed, which is yours to keep wherever you like.
Photographs are the exception: they are not in the export yet. If you need them out before that lands, say so in an email and it moves up the list.
Clearing data, or deleting the app
Clearing your data from Settings takes a backup first, without asking, and abandons the whole thing if that backup fails. There is no undo, so the backup is the net.
Deleting the app takes the data on that device with it. There is no copy on a server of ours to ask about, because there is no server. If you had sync on, the copy in your own iCloud is yours to remove from iCloud settings.
It will not install, or it will not open
Cadence needs macOS 15 or later or iOS 18 or later, and about 37 MB free. On the Mac it runs on Apple silicon and on Intel.
If the requirements are met and it still will not start, email the version and build from Settings, then About, along with which machine and which OS version, and it will get looked at.
Refunds
The sync unlock is bought through Apple, so refunds go through Apple rather than through this address. Ask at reportaproblem.apple.com with the Apple Account that paid.
The app itself is free, so there is nothing to refund on that side.