Okay, so, last week I accidentally dropped my phone and cracked the screen. The silver lining to this story, though, is that I now know how to safely transfer PAD saves between two separate Android devices of the same model.
Here's what to do if you have to replace your device for damage:
***YOUR REPLACEMENT DEVICE MUST BE THE SAME MODEL AS YOUR INITIAL DEVICE! IF YOU TRY THIS WITH A DIFFERENT DEVICE, YOU WILL ALMOST CERTAINLY BRICK YOUR DEVICE!***
1. Root both of your devices and flash a custom recovery (not custom ROM, which is something completely different). I personally use and recommend TWRP
2. On your original device (Phone A), boot into recovery and create a NANDroid backup and reboot
3. Copy the backup from step 2 to your PC (if you're using TWRP, go to ...\Android\ and copy the whole TWRP folder)
4. On your new device (Phone B), boot into recovery and create a NANDroid backup. The reason for this for your recovery to create the necessary folder structure for backups
5. Connect Phone B to your PC and navigate to ...\Android\TWRP\Backups
6. Open another window and navigate to your backup from Phone A, then go to ...\TWRP\Backups
7. You will see that there are two serial numbers; navigate one level deeper in each of the windows from steps 5 and 6
8. Copy the latest backup from Phone A to Phone B
9. Reboot Phone B into recovery
10. Restore the backup that you just copied
After that, Phone B should be running a mirrored image of Phone A, so you can do whatever you need to do with Phone A. In my case, what I needed to do was unroot and relock the bootloader and then return it to Verizon.