Drains in initial wash cycle, then dies immediately after rinse fill..

Hello, I am sure this may have been already answered but I would like fresh ideas here. Two months ago my old, p.o.s. Kenmore 70 Series wouldn't drain. Took pump hoses off and pulled out one of my children's socks. Problem solved. Now, same problem, but there is nothing in the pump or in the hoses. Drained water, put it all back together and ran a cycle. It filled, agitated and washed, drained no problem. Once it went to the rinse cycle, it filled up again and then stops and basically dies. I can turn the knob anywhere and it does nothing. I am thinking it is not the pump, unless the pump isn't really activated until the rinse cycle. It drains once no problem and I am assuming the pump is utilized in the initial drain in the wash cycle, but then stops completely following the rinse fill. Could this be a motor problem? Or a timer? I don't think its the lid switch, because the lid switch works fine in the wash cycle. Please help! Thank you kindly.