Fan always on

Hello, you are probably correct, if you have no codes for the coolant sensor or anything else to lead you in a different direction then it is probably a bad fan module. Below is the cooling fan wiring diagram. The fan is controlled by the ECM on Pin #50 highlighted in yellow, the ECM PWM (pulse width modulates) the control module. So, it will send an on/off signal on that wire depending on what speed it needs to be on,
Now on the other hand, if the check engine light is on and the ECM is down, or lost communication, some cars will turn the fan on all the time because it doesn't know if it should be on or not, so as a safety feature it just turns the fan on all the time. I will look to see if there are any simple tests you can do to determine what the fault could be.
What I would do is De-Pin the red/white wire out of the connector at the fan module and see if the fan still runs. So, it would be running without a command from the ECM. But it can be hard to De-pin connectors sometimes because you do not want to break the connector. I have added below the 2 fuses that run to the Fan Control Module.
You should at least check the connectors at the fan module for corrosion or any sign of damage to the circuits.

https://www.2carpros.com/articles/how-an-electric-cooling-fan-works

https://www.2carpros.com/articles/how-to-check-wiring

I will have to compare the diagrams, but did you try the de-pinning technique? Just to be sure it's not an input from a different sensor. Is it a 3.2L? All you put was 4 cylinder so I assumed it was a 2.0 and if it is 2.0 is it a Quattro.
On that other diagram in parentheses, it just means the location of the ECM. Here is the 2.0 and the 2.0 Quattro. See it's just giving to locations of the ECM.
I just added the de-pinning because it's the fastest way to know if the ECM is turning the fan on or something is burnt in the fan module and keeping it on,
I was curious, I see you on here a lot, do you have a scope too? Pico or snap on?

No, I mean the wire coming from the ECM to the fan module, Highlighted in blue here (red/white wire). This is the wire that has a pulse width modulated (so off/on very fast) signal from the ECM to the fan module. There are ways to take wires out of connectors without destroying them. If you look at the plug end that goes into the fan module, you'll notice a space where you can put a small pick or very small flat head screwdriver in, I will add pics, this is a good technique to learn because it can save a lot of time, you don't have to test temperature sensors etc, to see if they're the issue. Sometimes you have to pull the weather pack out of the back of a connector, depending on the type of connector it is. But be very easy on the connectors, don't force them apart. A lot of techs do this with communications issues because it can eliminate a whole side of a circuit and save a ton of time on testing.
You just have to see how the connector is put together and after a while you'll get good at it, and it will be less of a hassle.
So, if you take the wire out and the fan stops running, you know that it's an ECM input that is keeping the fans on, if it keeps running you know it's a bad fan module. If this all makes sense to you.

Also added a pic of de-pinning tool for example.

Is Fuse #11 labelled something different? Something like "Motronic Engine Control Module Power Supply Relay? And did the fan stay running with the car off before? Or is that a change? The two power supplies for the fans are an interesting circuit. Fuse#11(5amp) and Fuse#1(40amp) are setup in a strange way. The Motronic relay powers the ECM too.
Fuse #11 is energized by the "Motronic Engine Control Module Power Supply Relay" and Fuse #1 is energized by the "Battery Interrupt Igniter". Let me know if that's the Label for Fuse #11. If you cut the ECM control wire and now you have the fan on all the time, then it seems the fan circuit is always grounded without that control wire. We will have to look into the Relay and the battery Interrupt Igniter. Can you tell me which of the wires coming into Fan Module has power, it will be the red wire or the red/green wire? And obviously with the fans running all the time now it won't matter if you unplug the fan module, you should still have that power there that's keeping them on. Depending on which wire is still hot is the direction we'll take.

Okay, the battery Interrupt igniter is a relay that also powers the starter motor, so if the car will start it can't be that, strangely enough that relay is turned on by the Airbag module. Very weird design. I should have asked this to begin with, but this isn't a convertible, is it? And does this car have start/stop control?
You wouldn't believe what these diagrams look like.

Yes, I see the wiring connections, the red/white is grounded threw the ECM, PWM grounding to control fan speed, those are only two of the fuse panels, there's more, there's one in the right side of the luggage compartment, there's fuse panels under the left and right side dash, there's a relay/fuse carrier E-Box SB in left plenum chamber E-box, Fuse carrier 4 and 5, 3, 2, and 1,
Main Fuse panel SA on battery in luggage compartment. There's 9 pages of power distribution diagrams, And they have labels running all over the place,

So, this Motronic Engine Control Module Power Relay is on the Relay/Fuse Carrier E Box SB. It's controlled by the ECM. Below is from that Relay to Fuse #11 (Relay/Fuse Carrier E-Box in Left Plenum Chamber E-Box) to the Coolant Fan Control Module.
There can be a number of things keeping the fan on all the time,

Can you access the Fan control module with one of your scan tools? Or do a full system scan for codes.
You can also monitor the Coolant Temperature sensor in the Live Data and see if its within spec. Or if you have any codes in any other modules that can give us a little better direction here.
The Motronic Relay can't be the issue because it needs to be powered up because it powers the ignition coils and a ton of other things, it can't be the Battery Interrupt Igniter because that is turned on by the Airbag control module and powers up the starter motor, the secondary air injection pump, the Generator and Voltage regulator.
So, it has to be something else. Have you tried a new Fan module yet? We've ruled out a lot of things. Or something is causing the ECM to keep it on, I'm not one for throwing parts at a car ever. But in this case, I would try another fan module.

No of course it's not a burnt fuse, but I'm saying these two relays power up so many other things you would have a ton of other problems. You would have charging system issues, no power to the coils, no start conditions. Below is a diagram with both of those relays.

Wondering if that's keeping the fan on, this car has two or just one fan in front? And is this a diesel engine? AllData and ProDemand have completely different options for this car, I've been bouncing back and forth between the two.
There're even different listings for the cooling fans, 600watt, 800watt, 1000watt.
And it's not a Fuse on the battery, it's a type of relay.

Below are these the wire colors for the ambient temperature sensor?

Okay, so here's where we're at, I'm assuming you haven't tried a new fan module yet, but the fact that you're still getting a code for the ambient (Outside Air temperature sensor) is relevant. You had 5-volt reference at this Outside Air temperature sensor? On the green/white wire. The other wire is a shared ground with the Coolant Level sensor and something else, I don't know what, if you see the shared wire between the two sensors, it's a dotted line going into the "Vehicle Electrical System Control Module" means that wire is shared with other components. If you had 5-volts at the Outside Air temperature sensor. That means you either have a bad ground circuit for that sensor, another bad sensor (the new installed one), or something is going on with the VESCM (electrical control module mentioned above). But if you had a bad ground, you would probably be getting a code for the Coolant Level Sensor.
On the diagram below in orange is that wire you cut from the ECM, (red/white wire). I'm wondering about that splice between the two sensors and whatever else is shared on that circuit. Are you able to look at Live Data for any type of Body Control module? It looked like you had an Autel scan tool. If so, you should be able to look at live data for any module. I think you should also check the Coolant Level Sensor just to be sure it has 5-volts and ground, if it does. The issue is with this Outside Air temperature sensor. It's tied into the A/C system and for whatever reason it may be keeping the fans on, If the coolant level sensor isn't setting a code, then it should be ok and so should the VESCM. Because both those sensors are grounded through that module. I think you should trace that Outside Temperature Sensor wiring all the back, both circuits, ground and its 5-volt ref. You need to find that splice and see why its tripping that code. What is the exact code number? I want to see the setting criteria.