Oil can leak into the coolant and still have no water in the engine oil, just depends on the fault. The oil goes milky on the dip stick with coolant in the engine oil. Don't suppose you are loosing transmission fluid from the gearbox? Some auto boxes have oil coolers in the coolant rad which could be a source of oil in the coolant. Lack of thermostat should not cause this problem, I don't think.
Change the coolant with water from the tap and see how you go.
You will definitely know in a few days driving, if its worth new coolant/oil