Thanks to original poster here.
I didn't get the process at first, so I rewrote it with more detail below.
Note: If you have already started HuntCook on your NEW device, uninstall and reinstall it. DON'T start it again (yet).
1. Start HuntCook on OLD device.
2. If you haven't already set it up, it may ask you to allow access to your Google account.
You have to permit this, as this is where the backup will be stored.
3. In the HuntCook app, go to Menu->Settings->Data Backup.
4. Proceed to back up your data.
- You will need to assign a password. Make it a simple one unique to HuntCook - you don't want a data breach goofing up your real passwords.
- You will get a cryptic 16-character "Backup ID". Be sure to write this code down.
- You will also get the option to take a screenshot. You can do so but I don't know where it is stored.
5. On the NEW device, start HuntCook.
6. Tap "Restore data backup" button in the bottom right-hand corner.
(It may be called something else, I forget the exact button name.)
7. Enter the cryptic 16-digit "Backup ID" you previously wrote down. Don't enter the spaces.
8. Also enter your HuntCook backup password.
9. You get prompted two or three times to allow transfer of the data from your old device to the new device. Confirm the transfer.
HuntCook will then import the data and deactivate the game on the old device. Your new device is ready to go.
You will be warned several times that transferring the data will prevent the transferred gameplay from being played on the old device. However, the app is not deactivated, and you can start a new game on the old device.