My Maytag Atlantis 2005 washer begins the first spin cycle as soon as the first wash cycle ends. So, the spinning starts with a full load of soapy water. Isn't it supposed to drain all the way down before the first spin cycle starts?
Is it draining at all, but just not fully? Does it eventually pump the water out, or is the water just sitting there at the same level no matter what?
Washing machines have a pump under the tub that pumps the dirty water out. These pumps often have a little mesh filter in front of them, sometimes there's also a filter on the outlet where the drain hose connects on the back of the machine. If these screens get plugged, the pump will be very slow and may not have enough time to clear the water out between cycles. So, if the pump seems to be pumping, just not quickly enough, check for these filters and/or check the pump itself for obstruction.
If the pump isn't turning on at all and the machine is just sitting full of water, google your machine for a troubleshooting diagram or parts diagram. Often, the pump circuit will have sensors or switches in it, that prevent the pump from turning on unless certain conditions are met. For instance, some machines will not turn the pump on unless they can detect that the lid is latched shut. Other times there will be a water level sensor or other sensor that the pump relies on. Troubleshooting guides provided by manufacturers (or just detailed troubleshooting info on appliance repair forums or blogs) will usually point these out and tell you how to check them.