2001 Dodge Caravan Electrical/Starter Problems

There are two major problems which both seem to have started at the same time. The car's starter motor will not engage when the battery is fully charged, and all we get is some clicking form under the hood. This initially made me suspect the starter/starter solenoid.

However, since recharging the battery and leaving it in overnight, I came back in the morning to a very flat battery. Something drained it overnight, and regular clicking noises were coming from near the fusebox, and also near the alternator.

So in summary, the car won't start AND the battery suffers a parasitic drain. Both problems started at the same time. Any ideas?