If you didn't find any contamination from engine oil the next place I would check would be the transmission. I have seen the radiator fail at the core where the tranny line cooler is. If it has just started you might get lucky and it will only bleed off from the radiator side being that the cooling system still has pressure until it cools down.
You stated the car was new to you in the last two weeks maybe the previous owner tried flushing out the coolant and what you are seeing is the residual of it. From the symptoms that you have mentioned it sounds like you may have an electrical issue or a faulty sensor causing the fans to constantly run. I would first check the engine coolant temperature sensor on OBD2 its called an ECT. Its job is to measure engine coolant temp, it affects fuel delivery and idle speed as well as the ignition timing and egr. On fuel injected engines it works with the PCM to function as a carburetors "choke circuit"
I am not sure your level of experience so maybe this will help someone else if not you
Thermistors- purpose is to vary resistance according to temperature change inside is a semi-conductor crystal known as a negative temperature coefficient ( NTC thermistor ) Reference voltage is passed through a fixed resistor within the PCM & then through the NTC thermistor. As temperature increases -- resistance decreases and voltage signal decreases which pretty much means you get a voltage drop over the thermistor.
General temp & resistance parameters
-40*F = 100,000 ohms
+210*F = 2000 ohms
Again I can go on and on just trying to help tim