Thanks for your response!
I cleaned up some sludge buildup in the output pipe and around the water pump's output flap and then rinsed out the water pump. I could spray water in the input side and it would flow out of the output side. So there wasn't any object inside of it that I could tell. It didn't appear to be stopping the water from exiting the water pump. I could easily move the wheel inside the water pump by pushing my finger inside the output part of the pump and turn it.
I kept the front off the washer so I could check it out and watch it through a cycle. It doesn't leak when the water is going into the washer or through the spin cycle, but as soon as the spin cycle stops (and the water pump stops by design I believe) and the washer starts to fill up with water again then the water starts to leak out immediately from the water pump around the place where it hooks onto the engine shaft. It is a steady stream of water. There is no crack that I can see in the water pump which makes me think it is leaking out around the engine shaft. As soon as the water pump turns on again when the agitation cycle begins, there is no leak. It is fine again all the way through the agitation cycle and through the spin cycle (the water pump is on) until the spin cycle stops (and the water pump stops again). Once the spin cycle stops, then the leak begins again because the water pump stops. Do I need to replace the water pump or do you think it is something else.
One good thing is I don't seem to have the burn smell anymore. I don't know what I would have done to stop the smell. But it seems to be gone now!