The blower fan should not run when the climate control is switched off. You could have a problem with the pushbutton unit or less likely the motor speed control module located under the blower motor on a heat sink.

Mercedes used 2 manufacturers for the pushbutton unit in the W124. If it is a Bosch there is a 5 volt regulator in it that gets so hot that the solder eventually cracks. Also cracks develop on the switches and temperature control potentiometer connections.
The other more common unit is made by MK. It's failures are usually solder cracks on the switches and temperature potentiometer. Sometimes electrolytic capacitors.

To test the blower control module you could disconnect the yellow wire that is in cavity 3 of a 3 terminal connector behind the brake booster. That connector has red, black and yellow wires in it which go to the speed control module. The yellow wire is the control wire from the pushbutton unit. If it is removed from the connector and the fan still runs, then the module is bad.

For lubricating the blower bushings I have always used transmission fluid applied with a syringe. If the brushes and commutator are still in good shape it has always worked well for me. Avoid getting fluid on the brushes and commutator.