There are very few root methods that don't require a computer, and those that there are only work on some devices. But are you sure that you can't run adb and fastboot? They are not the same as regular USB file access. If you are able to follow Scotty's instructions and run the Moonshine S-Off then install a recovery I'd think that you'd not wipe the phone - it's unlocking the bootloader the official way that does that (Step 1 of post 2 of the rooting guide, the one that Scotty tells you to skip in post 1).
If you are using the same Google account on both phones can you not synch the passwords back to the new phone? I don't use this option, so don't have any direct experience, but I'd have expected it to work.
I don't actually own a DNA (I'm a Brit, so no Verizon here), so if you want people with direct experience of the device to advise on rooting options we could move this thread to the device forum. I could ask one of the moderators, or you could do it yourself: just use the "report" button and ask if the thread can be moved to "DNA All Things Root", that will send the request to the moderator team.