Without removing the transmission from the car, I would do the following:
-Put the transmission in neutral and remove the shift tower.
-Place a drip pan under the ****** and remove the drain plug.
-Put a gallon of mineral spirits into a pump-up garden sprayer and rinse the insides of the transmission thoroughly while rotating the gears by hand.
-Let it drain and allow the mineral spirits a day or so to air-dry completely before adding gear lube. If you prefer, use compressed air to help the drying process.
-Reinstall the bottom drain plug and while the top of the ****** is still open, remove the filler plug on the side of the ****** case and add your fresh gear lube directly into the open ****** case at the top, being sure to drench all the gears while moving them back and forth on the shafts to make sure new lube gets into all the crevices, bearings, and shafts until the level reaches the filler plug opening and starts to overflow it.
-Reinstall the filler plug and shift tower and you should be good.
Others may have their preferred method, but this seems to work fine.