Way back during the Dependable Care days, you could get by with oiling the brake package. It was not the same design as the Herrin platform machines (Performa, Atlantis, or anything with 1 belt)
As schaffer said, oil will allow the stator to slip, causing the tub to spin during agitate. The solution is to :
1. Spend money on a new stator
2. Take the stator off and sand the inner ring till it does not shine any more.
p.s. The stator is the outside cover with the 6 black screws. Unfortunately, it's under 200# pressure.
If you ain't got the tool, you ain;t fixin either one.