2001 Oldsmobile Aurora Running Hot

Cooling fans coming on? Any white smoke from tail pipe?

Hate to say but your the proud owner of a blown headgasket. But could be intake manifold gasket but that usually turns oil a chocolate shake look. Not many miles a fix and drive for few more years.

Sad to give such news in these times but at least you know and can save engine.

These motors are notorious for blowing head gaskets.
The easiest way to check:
Take off the cap to the surge tank. Have someone rev the motor while you look into the surge tank. If you see bubbles that are larger than a millimeter, then exhaust gas is getting into the coolant. (Blown head gasket)

White smoke will come out of the tail pipe only the first few minutes as the car warms up. This is because coolant has seeped into the exhaust system and it's burning out as the car warms.

But as the engine is running, no more coolant can get into the exhaust because of the high pressures in the cylinders. This is what makes the exhaust blow back into the coolant system.

The reason your car overheats is because that exhaust gas is creating air bubbles that will collect inside of you heater core and radiator. This stops the coolant from circulating correctly.

Unfortunately, NO ONE wants to rebuild these motors. (Requires a large investment in tools that can't be used on any other motor)

Your only option is to find a used motor or order a crate motor from a dealer. (Astronomical)

If you decide to replace the motor, make sure to have the mechanic look up a T.S.B. (Technical service bulletin) that was issued by the manufacture for your vehicle.

They require a special thermostat. (Why these motors blow their head gaskets)

If you replace the motor without locating this special thermostat, you will only blow that motor also.