Do you mean that the only way to kill the motor is to stall out the car or disconnect the battery? If you stall out the car, do the accessory functions stay on?
Or, if you turn off the key, do the "ignition on" features like the windshield wipers and the directional signals still function? Does the motor continue to run also?
The ignition switch is just a glorified circular switch - as you turn the key, it opens up different circuits. The "on" position sends power to the alternator and accessory circuits [wiper, directionals, brake lights]. The "start" position adds power from the battery to the starter motor.
Perhaps the internals of the switch are not functioning properly. The internals can be replaced and allow you to still use your current key. I've had to replace mine on my II-A twice in 19 years, once in a snowbank and once on a -15F morning. If I can do it, so can you
. The Haynes Manual or Land Rover Service Manual will help you identify which color wire goes on which connector of the switch. Helpfully, each connector has a letter designation on the switch itself.
The Series III has the ignition switch that includes the starter function, and is probably shared with other British Leyland products of that time. My MG Midget and MGB also need switch internals replaced.
Good luck,
Jeff