I have quite some experience on these old guys.
The original lube is dried and needs removed then replaced. I take out the old cable,usually it must be unhooked from the speedometer from under the dash. Pull it out, let it hang a C shape and spin with your fingers, if it jumps up and down some of the wires are broken, get new. Otherwise soak in solvent.
If you have cruise, then there are 2 cables, you must do both.
To clean the housing I spray with carb cleaner, then run a wire down it, make a cloth plug like cleaning a gun saturated with carb cleaner and pull thru. This gets out some of the old grease.
I use a thin light grease, or just oil when I replace. Oil is better than too thick of grease. Chassis grease is too thick. Sometimes special ball bearing grease or bicycle grease is good.
The grease in the cruise module and the speedometer head may also be old and thick. Same drill, carb cleaner and oil to thin it. This usually gets you a few years.
Rod