2000 Lincoln LS Relay/Cooling Fan

Your alternator wouldn't cause your car to overheat.
And your car only uses the fans to cool the radiator while you are sitting still. At speed, your radiator is cooled by air flowing through the front of your car.

When do you most often experience your overheating problem? And if you are overheating but your fans are only turning slowly, then there definitely is a problem. If you're familiar with a DVOM, then you can use that to test your fan relay and the motor windings of the fan. Also check the wiring/connectors for looseness/corrosion.

If you're experiencing overheating at speed, then this won't be your fans. You'll have to have your cooling system diagnosed for other issues. Since you've already replaced the therm, and water pump, you'll have to check your radiator/heater core/engine block for a blockage.

If your system checks out, then you'll be looking for a blown intake/head gasket.