Many times on Chrysler products of that vintage, the ignition switch will "bounce-back" just a 'fuzz' too far when the key is released immediately following cranking the starter until the engine is running. Sometimes the vehicle acts exactly as you have described. Sometimes the radio is inop as well. Try turning the switch forward a very small amount and see if this helps.
Sometimes the ignition switch just plain fails.
Take a test light and ground the clip, check the fuses for power supplied to them by touching the little "window" cavities on the fuse. (I usually do the test with the key on, and the lights turned on. I always test all the fuses in the box, heck - the tool is already hooked up.) Blown fuses have power at one "window", but not the other. If neither window "lights" the test light, there is no power to the circuit. (It's the ONLY way to quickly test fuses, you can be 'burnt' by visually looking at fuses, and this tells you more of what is going on, in just seconds!)
If the fuses in question do not have power, and the main feeds to the ignition switch do, the switch would not be sending power through.
Jeff Ruffing
ASE Master Auto Technician
Chrysler Master Auto Technician
65 Imperial Crown Convert
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