yes, the connector on the ignition switch may be burned. have someone check it for you.

Roy

Did you replace the ignition key lock cylinder (the metal part which the key fits into) or did you replace the ignition switch (the plastic part behind the key lock cylinder where the ignition wires plug into)?
Sounds like the problem is in the ignition switch.

To test your mechanics theory of it being the shifter cable, put the car in park and then try to push it to see if it rolls (if it does them it could be that cable is out of adjustment; thereby not allowing the gear selector to put the car in park). In this case the ignition could still be turned on slightly allowing the dash lights to stay on.
Also, shift to neutral (two clicks from park--don't pay attention the shift indicator--that way you're sure it's in neutral--on some cars the indicator is off just a little) to see if then you can push the car. If all of this is correct, a bad, misadjusted, stretched cable is not much of a possibility.