Sounds like the cylinder isn't lined up to the key way.
The ignition lock cylinder on a 2006 Chevy Cobalt is removed by popping off the steering column cover and inserting a screwdriver into the release hole. The cylinder can then be pulled from its housing and replaced.
With the cylinder removed there is no easy way to turn the switch it would normally operate to the "off" position.
The ignition is removed by taking down the kick panel below the steering wheel. They are a bear to get out. Make sure you have the whole cylinder assembly because they most often break apart when you remove them.
The ignition cylinder on a 2002 Hyundai is a quick repair. First the lower steering column will need to be removed so the lock cylinder can be accessed. Once off, loosen the mounting screws for the cylinder. Insert the key and push the tab in while turn the key slightly so the cylinder can be unlocked from the housing.
Steering wheel has to be removed. You need special tools.
If you are talking about the ignition lock cylinder, the steering wheel must be removed and the column disassembled to access a lock screw which holds the ignition lock cylinder in the column. No key is required for this (unlike all other vehicle manufactures). If you are talking about the door lock cylinder, the door panel must be removed, and the lock cylinder retaining clip must be removed (again no key is needed, which is the same for all manufacturers). If you are talking about the trunk lock cylinder, it is easily accessible from inside the trunk. Those are the only types of lock cylinders in this type of vehicle. There is no specific lock cylinder named key lock cylinder.
A dealer can look up the ignition codes to make a key if one needs to be replaced on a 1992 Olds 98. A locksmith can also help but the ignition cylinder has to be removed.
Your key switch contact inside the ignition switch cylinder may be stuck closed. You may need to replace the switch or the ignition key cylinder. On a Ford and GM, the contact is replaceable without the ignition switch, but requires the steering wheel removed to replace it. Not sure what is required for the Lebaron, but it is not a major task to do for a mechanic.
It is possible, would be a odd coincidence.
To be clear, this is the cylinder ignition lock we are talking about, not the ignition starter switch. If the tumblers are damaged or 'bad', the key may be able to be removed fromt the ignition lock while the car is running. Also, the key may simply not go into the ignition lock anymore. On most cars the cylinder ignition lock is a separate item and can be replaced for a moderate price. However, this is not something for a novice to work on since this is just one of many interconnected parts. Messing with a ignition system may make your vehicle unable to simply start again.
My 1995 Ford Explorer has also done that occasionally ( although I have removed the ignition key as soon as I open the drivers door , the key in ignition chime starts " going " ) What I do is put the ignition key back in and then remove it again and the chime will stop Give that a try
If the ignition cylinder is broken; and causing the vehicle not to shut off, that is an easy fix. With the key in the ignition and the cylinder turned to the "run" position. You would need to insert a rather thin tool (such as a Phillips screwdriver, a pick tool, or an Allen key), into the small opening in the bottom of the steering column. Most models have the release pin exposed, if this isn't the case, just remove the bottom portion of the steering column cover (remove tilt adjustment rod, and five screws - there is one near the dash on the left - it goes horizontally). There will be a silver button in the bottom of the cylinder. If you push the button very hard, and pull the cylinder to the right, the entire assembly will come out. If the vehicle won't shut off with the key removed, it could very likely be the ignition module, which should also be on the steering column. However; I am afraid that despite my experience with town cars, I have not had to replace any.