1997 Mercury Cougar Ignition wont turn off

Sorry for the late reply. Check primary voltage to coil pack, red wire with lite green tracer, that goes hot with the key on. If that wire is hot, either it is shorted to voltage or possibly something wrong with ignition switch. Another possibility is the vehicle is dieseling, too much cylinder compression or maybe a hotspot in the cylinder.

Just a thought on the battery running down, I'd have to make sure battery has a full charge, connections are all good and charging system is working ok.
Another possibility is too much battery drain with everything off. In that case, check for a light left on, somewhere, interior--under the hood--in the trunk. Best time to check is in the dark. After that, pull/replace fuses one at a time, when you pull the fuse on the faulty circuit, the drain returns to normal, then you check everything on that fuse circuit. Sometimes a faulty alternator may let the battery drain back through the alternator.