will bad coil cause battery to drain
I doubt it. With the ignition off there should not be any power going to the coil.
Yes, I believe yours is always hot to one side of coil and computer turns on and off the negative side, so if your is grounding, it will stay on. Check with ignition switch turned off and see if you get 12 volts at the coil. You shouldn't, if it's working correctly
I have had a dozen or so bad coils on my Triton engines and never a fault like that.
Google on You Tube "parasitic draw test". tons of instructions on how to find it with a multimeter. Could be alternator, or a zillion other things.
Looking for a Used Grand Marquis in your area?
CarGurus has 2,028 nationwide Grand Marquis listings starting at $1,500.
Search Mercury Grand Marquis Questions
Mercury Grand Marquis Experts