Yes, indeed, something could be loose in the shifter housing. Lots of things get loose in there, in my opinion.

The neutral start portion of the gearshift switch is necessary for the shift lock. But-- the car would not crank if that part were failing.

The reverse lights portion is not involved.

But the brake lights are definitely involved. The power to retract that lock solenoid comes directly from the brake pedal switch; does not pass through the bulb-out sensor. But if that power is there (i.e. the brake lights are coming on), the shift lock relay has to pass it to the solenoid. The shift lock relay must get its ground through the starter solenoid via the neutral start switch, and its power from fuse 12.

The suggestion to check those lights is helpful in most cases as a shortcut to actually tracing the juice with a test lamp. Could be your snowflake is just burned out, I suppose, but I'd start by testing at the fuse panel. Then maybe you can see what's going on under the shifter cover...

BTW, here's how I dealt with the failing reverse light switch: Cheap fix for the backup light switch on automatic

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Art Benstein near Baltimore