Don't forget those little plastic rollers. Mine were binding.
I used carb cleaner to clean, and white lithium spray grease to lube. Huge improvement, and no problems since.
I have nothing against the ATF, but have always used lithium for this application on various cars and had no problems. My reasoning is that the grease is thicker, so perhaps it will stay put longer. The spray form is also very convenient to use.
Best of luck.
1986 300E 5-Speed 240k mi.