Transmission problems are common in this model, often as a result of a failed front pump. The pump loses its ability to create hydraulic pressure, which an automatic transmission needs to function. A sign of front pump failure can be a whine type noise.
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Repairpal verified 2006 Ford Crown Victoria problems
The engine may overheat at idle when the AC is on. A new, upgraded fan motor and resistor assembly should be installed.
The composite (plastic) intake manifold may crack near the thermostat housing and cause a coolant leak. Ford released an updated manifold that was reinforced to prevent a recurrence. No recall was issued for this problem but Ford did extend the warranty to seven years on some models from the date of purchase.
The throttle position sensor (TPS) may fail. Common symptoms of a defective TPS can be rough idle, hard starting, hesitation on acceleration, poor fuel economy, and stalling.