Frost buildup in ice-dispenser tube

Is it a leak or a drip? A leak means you have continuous running water and a drip is more inconsistant.

I would suspect that you have a drip, that is caused by your cooling tube having frost or ice buildup in it, which is quite common for some units. The fix that seems to work the best is to defrost your unit, making sure the cooling tube is completely free of ice and frost by blowing hot air from a hair dryer down it. Then cutting a piece of bare electrical (solid core) wire that will fit down the tube but not extend past the opening. Then you form a hook and put it around your defrost heater.

That way the heat from the defrost heater will be transferred down the wire and keep the tube free of ice and frost.

1)The door gasket on the freezer door helps keep the cold air in and the warm air out. If the gasket is dirty or damaged, it may prevent the door from closing properly, which can lead to frost buildup on your freezer walls. To make sure the door is sealing, slip a dollar bill or a sheet of paper between the gasket and the freezer and then close the door. If the bill or paper pulls free with little or no resistance, the gasket isn't sealing properly. Wipe the gasket with a damp cloth to remove dirt, grease and other residue. If the gasket is damaged, replace it with a new one. Sometimes frost can accumulate on the back panel of your freezer if the door is open too long or too often. When you open the door, you let warm, humid air inside, and this condenses and freezes. Make sure there are no packages preventing the door from fully closing, and limit how often you open the door by removing what you need all at once.
2)Ventilation is important because it ensures your freezer stays cold. If something inside the freezer blocks the internal vents, the freezer may become warm, causing condensation to form on the walls, and eventually leading to frost buildup. When loading your freezer, make sure there are no items resting against the rear wall of the freezer and space the contents out as much as possible to improve ventilation.
3)Side-by-side refrigerators with a built-in ice maker may accumulate frost if the flap on the ice dispenser doesn't close properly. This allows warm air to enter the freezer side, condense and freeze, causing frost buildup over time. Open the freezer door and examine the flap inside the ice chute. It should be flush against the housing. If it is not, empty the ice bin and make sure there is no ice or debris preventing the flap from closing.
4)If your freezer door closes properly and nothing is keeping it open, frost buildup on the back panel may indicate a problem with the freezer's self-defrost system. The self-defrost system turns off the compressor for a short time every six to 12 hours so it can turn on the defrost heater. This melts any frost buildup on the freezer coils, and the water drains to the pan beneath the freezer. If this component fails, frost continues to build. Move the contents to another freezer and turn your freezer off for 24 to 48 hours to manually defrost the interior. If frost builds up again, contact a qualified appliance repair technician to inspect the defrost timer and repair the problem.

Probably the trap door where ice dispenses out is not sealing tightly causing frost buildup

Back bottom right hand side.

Replace the defrost clock.

How old is "OLDER" ? If there is no frost buildup on the back wall of the freezer you could have one of two problems. On older units the compressor wears out with years of service and no longer pumps efficiently. Or it could possibly be low on refrigerant charge. ether one of these two problems is a sign that is time to update to new unit.

If you have frost buildup on the back wall of the freezer section you then would have a problem in the defrost circuit attributed to the timer, defrost heaters or defrost thermostat.

I would look to the condenser where the heat is removed from the refrigeration system for your problem. Check and make sure the condenser fan operates, check the compressor for operation. If the condenser is plugged with dust it will cause the compressor to cycle on its thermal overload protector. Normally in back there is a removable panel to access the condenser and compressor area. make sure you reinstall the cover after you are done. It is critical to proper operation and cooling of the system.

wow you have power to the unit what about the temp control ? is it turned up?? if so is it closed(calling for cooling?) that is the place to start you can jumper it after yo unplug it to see if you dont have a meter

  It is not defrosting.  You have to remove back wall of freezer where frost is and take hairdryer to melt away frost.  You need to check defrost heater.  This is on bottom of evaporator coils.  It has wires going to it on both sides.  This has to be checked for continuity.  If ok on top of coils is a round unit called a defrost terminator.  This needs to be checked for continuity also.  This must be done before you defrost coils.  It should have continuity when its cold. 
  If these are ok you probably have a bad timer or defrost board. 

Is the trap door flap closing properly where the ice comes through the door? If open, warm air would be coming in frosting and maybe iceing things up