Slip the belts off and turn the transmission pulley counterclockwise (looking from underneath the machine). This should cause the agitator to move. Spin the pulley clockwise (opposite direction) and it should climb up the helical shaft, lock and spin the inner tub (you'll have to overcome some resistance from the internal brake spring tension). If it doesn't spin the tub, it's probably worn or broken clutch washers inside the transmission..The machine just agitates when the dial is in the spin mode instead of spinning.
The oil - would that be a transmission seal gone bad?
I don't want to get a new machine. This one has served us well for 38 years.