I've got an 83 300 diesel and when I bought the car I was able to remove the key completely and the car ran. It wasn't a vacuum leak, but was due to a broken-then-modified ignition switch, so you may have an internal mechanical problem with the ignition switch.

I'd do a bit more toubleshooting or collect some data points:

When the key is in the ignition and it's running normally, what happens with the open door? Do the same lights come on, or is that only when the key is out?

When the key is out and the engine running and the door open, do you get the "key still in ignition" buzzer, like the sensor for the key is broken? If the engine isn't running, the key is in and the door open, do you get the buzzer? I'd try to do things that can help diagnose a mechanical or electrical problem.

If you don't close the door, but only push on the sensor/switch, does that shut the engine down? That might mean there's another electrical issue. If it only does it when you close the door, then maybe there's a vibration/motion that causes the ignition switch to click closed.

And welcome to the forum.