Can rain damage a device so it drains more power?
Yes, anything can happen.

Even pure water is somewhat conductive and low power devices like an E-reader rely on the internal signal being completely isolated from one another. Even a small amount of water can compromise this isolation and then anything can happen. Without the proper isolation, currents can "leak" and that can drain a battery.

Also water in combination with the voltages that are constantly present in the device (since it has a battery) causes corrosion so contacts break and unwanted paths for currents to flow can form.


  I would assume water would make the device unusable
  That depends on the amount of water and how long it is present in the device.


My guess is that there is still some water left in the E-reader. You could try putting the E-reader in a closed container with uncooked rice. Rice attracts water. This is a fairly safe method to try.

Less safe is putting it in an oven at low temperature (60 Celcius or lower) for a couple of hours. You have to trust the oven though, that it will NOT heat above 60 C I mean.

Edit: Pro tip: remove the battery before doing any of this!