Often times when a fridge has a water drip as you describe, the cause is simply a clogged defrost drain. Nothing is broken, just clogged. Depending on what this is clogged with you may or may not be able to easily perform this type of refrigerator repair. If its clogged with ice, and not food debris, unplugging unit, opening both doors, and blowing fan for 12 hours at freezer will result in a fix. If the bottom of your freezer builds with ice that again leaks onto kitchen floor, the food debris inside the tubing needs to be cleared out by a repair service.

It is extremely hard if not nearly impossible for defrost melt water to travel from the evaporator located above the freezer section to the floor of the refrigerator. I checked three different sources and none show an image of this evaporator case support, the part that GE tech said you required. However, assuming it supports the evaporator coil as the name implies, and knowing how difficult this model is to work with even for an expert refrigerator repair service, I would highly recommend you call around for local services to have someone at least inspect and diagnose your unusual problem.