One common reason for water leaking under the produce drawers in GE freezer refrigerators is the self-defrosting system. The automatic defrost system on a refrigerator is meant to keep the evaporator coil clear of frost buildup by using a small heater to melt it away every few hours. The water from the melted frost goes into a drain and flows down to a catch pan under the unit, where it evaporates. If the drain or tubing becomes clogged, water spills down into the fresh food compartment and pools under the produce drawers. In a bottom freezer model, the defrost system works the same way but the water would back up in the bottom of the freezer and form ice in the lower storage area.
Your GE refrigerator may be a model with door heaters and a switch that can be turned on to cut down on electrical usage by turning the heaters off. These door-frame heaters discourage condensation in the fresh food compartment. If they are turned off and there is a high level of humidity in the refrigerator it may form on the ceiling or walls of the compartment and run underneath the produce drawers. Turn the energy saver feature off so the heaters will work and see whether the problem goes away.
Your refrigerator might have dirty, damaged or otherwise poorly sealing gaskets on the door. The strip of rubber that makes the door airtight is called the door gasket. Check the condition of the entire gasket, looking for damage or excessive dirt. Clean or repair the gasket to seal out moisture that could be causing condensation to form inside the compartment and running under the produce drawers.
If your GE refrigerator has a water dispenser built through the door, there is a source of water feeding it. The dispenser’s water supply could be leaking though a connection or a damaged piece of tubing. Check the entire path of water from the inlet to the tap to see whether the dispenser is dripping water into the fresh food compartment where it could collect under the produce drawers.