Headlights turn off my car?

Hello, this sounds like a strange case, there must be either a relay or some control module that is staying on or has a short to ground and causing it to stay on. When some relays stay on even when the control for it has shut off, it can cause power to back feed into other circuits. It's very odd with the key out that everything stays on. I assume the dash lights are staying on with the headlights. And with the headlights off during the day, does the vehicle shut off normally?
This might be something that needs to go to a shop to be diagnosed. But just to check a couple things. If you pull this Fuse 48 Headlight 30A in the engine compartment fuse panel. Does this cause the engine to shut off?
The Fuse 48 goes to the Headlamp switch which in turn goes to the Anti-Theft system as well as the Integrated Control Unit which is built into the dash fuse panel. So, there might be something on that circuit which is shorted in some way. And the Headlamp switch turns power off to some of these circuits.
But most times with vehicles that do not shut off there is an alternate power being provided through a bad relay or a module not shutting down. There isn't many other things that will keep the entire car powered up.
But pull that fuse and let's see if that's the circuit where something might be staying on.