The water inlet valve on the back of the unit where the water supply comes in can have a slow leak and will cause ice to buildup and block the fill tube. Disconnect the ice maker line from the valve and watch to see if it has a slow drip. It doesn’t take much to eventually block the fill tube.
The valve can leak because of sediment in the water preventing it from closing or low water pressure.


Your drain tube may be stopped up with ice at the upper end because it drains too slow because it's stopped up at the lower end in the evaporator pan under the unit at the floor. It can get dust and mold in it. Once you get the ice out at the top a little air pressure will usually clear it out. Flush it out with hot water and clorox may help. Make sure it drains quick enough. The drain should be located below the evaporator coils on the lower back of the freezer.

The valve on the back of the whole fridge is the one that may be leaking.

You really don't have to get to the bottom of the drain tube. If it is draining slow or not at all in the freezer, you can usually unstop it from there once you get the ice out by using a turkey baster etc. to blow air through it.

Can you explain how the slow leak in the valve causes ice buildup and ends up blocking the fill tube? I'm a little lost as to the chronology of events that happen.
Does the blocking of the fill tube then make the ice maker nonfunction? What is causing water to come out and end up on the floor. Sorry for all of the questions ...I'm just trying to picture it all.