Hello, I don't think the vehicle requires being in park to shut off -- perhaps to get the key out, yes, but not shut the engine off.
You did not indicate which engine your 300 is equipped with.
The shift lever assembly communicates with the transmission control module to determine shift lever position. There does not appear to be a separate sensor for this, which would seem to indicate the shift lever has such functionality within it.
Therefore, hands on diagnostic testing using a diagnostic tool that can monitor shift lever position as read by the transmission control module is needed, so you can figure out what's going on and what needs to be done to fix it. It could be a shift lever issue or a wiring issuer perhaps.
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