Kenmore washer goes thru all the cycles at the two spin cycles it will pump the water out but it won.t go into the cycle if you grab ahold of the drum and manually give it a start it will the spin out what is it CLUTCH MAYBE

Pull the washer out a little awy from the wall,tilt it back so the top rests on the wall,feel around behind the motor,it sounds like the tranny is going and you might find oil around near the back of the motor where the coupling is,also check the coupling,it's behind the motor,it hits the tranny to and when the motor spins it spins the tranny,i've seen the middle of the coupling wear out just catching and if you help it out it will spin but it sounds like the transmission is bad,it's not hard to change it out.send the model number,i can look up your machine,tell you how much a new tranny is and tell you how to change it out,or replace the washer if it's old and put the money into a new washer