cooling fan wont cycle

I have a 99 grand marquis ls and when it reaches operating temp the fan will kick on low speed, then several minutes later it will kick on high spped and will not cycle at all...if I shut the car off and immediately restart it fan will be off for 5 minutes and it kicks on again but wont shut off....coolant was nasty from previous owner but has since been changed. This happen with air off . I swapped fan relays but still does it. This car has no fan control module like the newer ones.

any ideas ? coolant temp sensor ? if so where is it, all I found was the coolant sender for the gauge passenger side of the alternator.

temp gauge is always in the middle when it reaches operating temperature

also was going to replace fan...anyone know what years are interchangeable ?

thanks