I have been through this.

On my '06 3.0, both elec aux fan and clutch fan failed. Car overheated at idle.
You really need both fans working...replace your Aux fan...I went with the TYC brand with lifetime warranty...It was 1/3 the cost of Genuine.

The clutch fan should lock up when the radiator gets hot...not just engine. But if you can stop it with a small roll of newspaper, then you should replace it.

Don't buy the Meyle brand fan clutch unless you want your car to sound like a bus all the time. Behr / Sachs is a good brand.

BTW - Neither fan failure will trip a code...