This is a highly suspicious estimate, either they are deliberately trying to screw you, or passively by suggesting replacement of multiple components instead of a few minutes labor to determine which one needs replaced.
THEY have the wiring diagram and hands-on to check things. Remotely, I have neither. Anyone have the wiring diagram for this? I just noticed that ebescohost site changed all these wiring diagrams around and is a pain to use now, but I did find one and from what I see, there's just 3 relays (high fan, low fan, and "fan control"), grounded through the PCM. These 3 relays are in the battery junction (aka Power Distribution) box under the hood, should be on a diagram in the owner's manual. For my '14 it's on page 303 of the PDF manual.
Does the fan keep running until the battery is drained or are you possibly just underestimating how long the fan should be running when the engine shuts off and pulling a fuse or relay to stop it? Is this a definite change in its behavior known because you owned it last year when it was warmer out and it didn't run so long after engine off back then?
The circuit diagram I'm looking at does not show where the PCM gets the temperature readings (which makes it incomplete but whatever) so some temperature sensor connected to the PCM is probably involved... one would think it's the coolant sensor which you should be able to get a temp reading from using an OBDII scan tool capable of live data.
There are other sensor triggers too, like the IAT sensor and A/C pressure sensor but AFAIK the IAT is not involved with keeping the fan on when the engine is off, couldn't be really because it's not drawing air in when the engine is off. You should be able to compare the running state temperature and whether it is dropping and where it is when the fan shuts off.
As far as the relays, you can pull the relay, get the pinout for them, and pin #1 on all 3 is the one the PCM grounds to run the fans. If the PCM is activating the relay then there will be 12V on the wire coming from the junction box socket corresponding to pin 1 on each relay. Two of the three relays are smaller and you could try swapping them for another two relays of the same size in the same power distribution box.
If you don't want to DIY this much, I would at least get a second opinion and forever after, avoid that shop. There does not appear to be any "fan control circuit" in a modular way that they could replace it (unless talking about a wiring harness that a mouse chewed up or something and it's grounding the fan to keep it on but this would be a new piece-of-wire $110 repair including labor), and a fan that is running okay just doing so when it shouldn't, is not a sign that fan itself needs replaced. It is obviously PCM controlled not a sensor built into the fan itself.
Now this is for my '14, has this changed on a '16? I don't see any reason it would have, PCM control would be the more evolved, final design for something like this, would make no sense to de-integrate a function already built into something else.
