I've done this successfully on at least three cars.

Remove the shifter and disassemble it. There's a circlip and a spring under the knob -- take those off and the concentric shafts slide apart. Coat the inner shaft with bearing grease and put everything back together. Sometimes the spring weakens with age, and shimming it with an additional washer or two can also help stop any buzzing or rattling.

Eric is quite right about the heater drain tube. You will be able to remove the shifter by lifting the carpet from the rear. You don't have to pull it out from under the heater box.