Can a bad starter start the car but still drain the battery over night?

Not likely. The starter would have to stay engaged to drain the battery and that you would hear loud and clear. Check for any light that is on. Dome, glove box, trunk, back-up, under hood, etc. It can also be a stuck relay. Lastly it can be a dead cell in the battery which would require the battery be replaced. If the battery is over 3 years old or has been run down a couple of times it is more than likely your problem.