If the shift lever is moving in to neutral, you should be in neutral and it should roll. The override thing is for cars like your 2002, where it has smartkey (electronic ignition switch).
As a test put it in reverse and see if the backup lights come on.
When the tow driver moved it, I think he flipped a catch which would keep the shifter from moving without the key being turned in the ignition.
My suggestion would be to put a battery in it and go from there. Or at least jump start it and see if it will go under it's own power.
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