Engine does not turn off with key

Does the car have an alarm or any other aftermarket electrical accessories installed? This sounds like a short to power concern.

Okay, very good! What happens with these Mustangs is that the bussed electrical center (fuse box) at the right front of the engine compartment (picture) has issues with pin fit and thus leads to burned terminals and wiring that will melt to each other and create shorts to power, shorts to ground etc. Very common so you're going to want to loosen the fuse box, I believe there are connectors held on by a 10mm bolt, flip it upside down for inspection and more than likely you will find melted wires! If the wiring is melted, most of the time you can repair them without having to replace the wire harness. If the fuse box is damaged, you can check with the dealer or go to your local salvage yard to find a good one ( that's what I would do). Let me know what you find!

Post a picture if you can.

Just like I have seen several times! This happens usually when the fan starts to go out and draws excessive amperage. You should pull back the wire loom and look at the wiring really good! Your problem is a short to power and this is the main suspect! At a minimum you need to replace the cooling fan assembly and tighten/clean the pin in the connector. You're lucky to catch it early because I've seen harnesses completely burnt and melted.

Very very possible. It's the first place that I would look and also a way to eliminate or verify this as a source of your short to power concern.