Dan O. is the appliance repair tech in here, and he'll be able to advise you better than I can. I suspect that the problem is that the drain hole in your evaporator pan is clogged up and needs to be cleared. It's a fairly common problem on frost free fridges. But, you're saying that water is collecting in the fresh food compartment of your fridge (presumably under the freezer compartment) and that could be because the water to overflowing the drain pan.
Here's what happens to the melt water on my GE fridges. Different fridges are designed differently so your fridge may have the parts I'll be talking about built into the partition between the fresh food section and freezer.
Just like your fridge, my GE fridges have a "drain pan" under their evaporator coils. When the defrost heater melts the frost off the evaporator coil, the melt water drips onto that drain pan. At the bottom of the drain pan will be a drain hole. The water drains into that drain hole and then flows through the partition between the fresh food section and the freezer to the back of the fridge. Once there, it drips out a drain hole in the roof of the fresh food section into a little cup (or funnel, kinda) high up on the back wall of the fresh food compartment. There is a tube connected to that cup or funnel that carries the melt water through the back wall of the fresh food compartment and then down to a receiving pan either above the warm compressor or around it where the melt water re-evaporates back into the air. If you use a mirror and flashlight, you should be able to see that tube between the back of the fridge and the black condenser coil at the back of your fridge.
If there's a hole in the roof of the fresh food compartment that your defrost melt water is coming out of (the one that you pushed the stiff 1/4 inch tube into), then you're missing a part. That is, the part that's supposed to catch that water and direct it into the tube that carries it down the back of the fridge to the receiving pan above or around the compressor.
If water is puddling up on your evaporator drain pan, it's probable that the evaporator pan drain is simply clogged up with food and stuff that falls through the openings meant for air flow, and gets washed down into that drain. This is a fairly common problem with frost free fridges. Appliance technicians will simply use a piece of wire to do that. Don't use a stiff wire; use wire that has thin strands of copper inside it for maximum flexibility so you can maneuver the wire so it turns horizontal and goes toward the drain hole in the roof of the fresh food compartment as soon as it goes through the evaporator pan drain hole. In fact, you might even consider putting that wire in a cordless drill and using it like a plumber's snake.