It's really hard to tell what is happening. The gears in the reduction box cannot be adjusted, and the worm is spring loaded. If you only notice it when cold then yes, I suspect it could be the grease getting thick. If you are handy, it's easy to clean off the factory grease and re-lube the worm and main gear. The gears in the reduction box of the motors are nylon and don't need any lube. The dec gear is the easiest to do. Just put the tube at its lowest point - at it's lower stop and unthread the dec lock knob. I don't remember is the 12" has a physical stop so you may need to lay a folded towel between the forks for the tube to rest against. Then remove the three small screws holding on the cover on and remove the cover. You can then clean and relube the gear without removing the scope from the tipod.
If you want to see what's happening, you can leave the cover off and put the knob back in place and run the drive.
One important thing is when the knob and cover are off it is possible for the main gear to be removed from the clutch assembly and its VERY easy to lose the little spring that holds the worm against the gear.
Did this problem just start recently?