2003 Pontiac Grand Prix

Engine Cooling problem
2003 Pontiac Grand Prix 6 cyl Front Wheel Drive 120000 miles

The radiator sprang a leak while driving on level road, engine overheated and maxed out temp gauage Only drove car for 1/2 mile before shuting engine down and getting a tow. Replaced radiator, pressure cap, thermostat, water pump, new hoses and replaced lost DexCool. Warmed up engine and Temp guage seems to hold at around 200 degrees and not coolant leaks. Cooling fans did not seem to be working as car sat there steaming after radiator blew out. Tested fans with a jumper to battery and they work but they don't want to come on with engine running and I switch blower fan to #5 with AC on. Have not driven car yet.
About 2 weeks before the overheating happened, the
heater blower went out and will only work on position 5
but only when I first start up the car. From what I have heard, the engine switch may be at fault for the blower problem but have not had that replaced yet. Could a problem with the switch be causing the cooling fans problem or could the problem be with the temperature
temp. Sensor or a relay?
Thanks for any help