How to fix the drain in our refrigerator

Here's an outside the box idea (which assumes that your main problem is that it wakes you up):

Get one of those timer plugs like they have for holiday lights and let it "unplug" your fridge while you sleep. You'll need a beefy one to handle the compressor spike when it kicks on (check the ratings on the timer plugs and compare it to the max rating of your fridge). Since the fridge door won't be opened, your food should be fine, though I'd test it the first night to be sure with a fridge thermometer. If your fridge is still cold in the morning, you might have a solution!

From GE knowledgebase

Refrigerator - Icemaker Leaking or Dripping Water

If your icemaker is leaking water into your ice bucket or onto your floor, several things could be the source of the problem. Check the following items:

• Be sure the water line fittings are secure.
• Check the water valve for leaks.
• See if water is leaking from the fill tube extension.
• See if the defrost water drain is cracked or overflowing.
• Check for water on the exterior of the refrigerator case.
• See if the refrigerator is leveled. How to Level a Refrigerator.

Note: If the icemaker is leaking water onto the floor of your home, turn off the water supply to your icemaker. This line is usually located under the kitchen sink. Locate the cold water line under the sink, and look for a small 1/4" copper line attached to it. Where these two lines meet, there will be a saddle valve. Sticking out from this valve will be a post with a wire running through it. This is the shut-off valve. Turn this valve in a clockwise manner until it stops. This will shut off the water to the icemaker. If the water line is not connected under the sink, it will be in the basement or crawl space.

I would check items 4 & 6.

For a clogged defrost drain GE says

Top freezer refrigerators:

Water under the vegetable pans is the usual evidence of a clogged defrost drain tube. Most top freezer models require some disassembly to access the defrost drain tube. We recommend that you contact service. If you would like to schedule a service appointment, please contact GE Consumer Service at 1-800-432-2737 or schedule a service appointment on-line.

Which isn't much help if you want to fix it yourself.

Here is an image of the cabinet assembly

And the unit parts (notice the drain tube is circled).

Images from Appliance Parts Pros.com

WARNING: If you are going to take the thing apart, make sure you unplug it first.