Add to that a sticky shifter.

Sounds funny, but I've come across shifters that had such thickened grease in them that there was fierce resistance to deliberate shifting, yet the thing would slip, slowly, as the surfaces sheared the viscous goo.

To fix this, shift into the biggest cog, then move the shift lever forward withe the wheels stationary. Now you'll have a lot of cable slack, and can easily disassemble the cover plate on the shifter. You don't have to remove the lever, but need to get some light motor oil into the center pivot of the lever, then oil the washers and cover plate and re-assemble.

With some oil in the shifter, the grease layer will free up, and the frictional characteristics will change from a sludgy "dynamic" friction to a pure "static" friction, and the lever should slip no more.
I would avoid any sort of ptfe teflon lubricants on this part, since it works by friction.