Battery drain fix and Non-stop vibration, or no vibration cause.
My Z was non-stop vibrating whenever the phone got a current, and with other Z's the vibration doesn't work at all.
There is the possibility that the vibrator itself is broken somehow, but in my case with my non-stop vibrating Xperia Z, I found that when I opened up my Xperia Z and pressed on the connector on the vibrator, it stopped vibrating. It always stopped when I pressed it, sometimes still being quiet after I have released the pressure I put on the connector, but the slightest bump would cause to vibrate it again.
I guess it's likely that the connector is broken, whether it's the connector of the vibrator or the connector on the phone, I can't say.
Anyway, any vibration issues don't have anything to do with the mainboard or other functionalities of the phone.
Considering the drained battery fix, if you're out of warranty, it's best to just remove the back cover, take the battery out and plug in a new one. It costs around 30 dollars. Or you could revive it at your local phone shop.
If you still have warranty and you need to recover important data, remove the cover, unplug the battery connector and connect a 2nd battery. Boot the phone, make a backup, and reconnect the original battery again. Clean the back cover of it old adhesive, attach the new adhesive (few bucks for OEM stuff), and send it back. There are no warranty void stickers, so it's unlikely they will be able to tell.