It sounds like something is blocking the drain system. Even a quarter will sit in the hole from the tub to the pump and block it. During agitation, when the pump is operating backward and pulling air in, the quarter will rise enough to allow the air in and might move around in the bottom of the tub, but when the washer starts to spin, the water carries the quarter back to the opening and it blocks the draining. Of course, something else can be clogging the hose or pump. Sometimes a small sock or a wad of lint will keep the machine from draining properly.
The reason the timer keeps going into a new cycle is that there is enough water left to satisfy the water level switch so the timer keeps advancing into the next cycle. This was a safety system of sorts in case a water valve did not close properly and water kept dripping or dribbling into the washer. Once enough water had leaked in to satisfy the water level switch, the machine would advance into wash and then spin out the tub of water without having the machine fill up and flood over the top and onto the floor. When it would hit the deep rinse cycle, it would fill and spin and sit there waiting for the dripping water to refill the machine. I thnk that once you take the hose off the bottom of the tub and from the top of the pump, you will be able to see what is blocking the water and be able to remove it without removing the outer tub cover and going fishing. Unless you have been smelling a burning belt, the pump is probably turning and the blockage is in the outer tub. Best of luck, Tom