Battery becoming very hot

That's interesting on a couple of points.
Did you look at the ammeter when you did these tests?
^^This will be the big clue. Which direction the ammeter moved, if at all?

As far as the voltage readings go.
key off shows that the lighter is hot all the time. That's correct.
Both tests show drops with for the load that suggest the battery needs to sit overnight on a slow charge like Tooljunkie mentioned.