Sounds like the water may be from the defrost cycle.
Remove the covers at the back in the freezer and check that the drain under the evaporator coils is clear
Sounds like drain tube is plugged so that the water from the defrost cycle cannot drain into the drain pan.

I would check that it is clear.
If plugged or partially plugged a turkey baster with warm water, an air compressor with low pressure or a wire can often be used to clear it.

Also check that it is not plugged with dust etc. at the drain pan end.
You might as well clean the condenser/compressor fan and the condenser coils while you are down there.

If it looks like stuff is growing in the tube I would flush it with a 10% bleach solution. This should kill the stuff and let you flush it out. You may have to do this several times, let it sit for 15 minutes or so between flushes.

Be careful that the drip tray does not overflow and empty the drip tray after you are done.

Be sure to unplug the unit while doing this.

The valve looks closed to me.
Usually if the handle is 90 degrees to the valve it is off.
If inline with the water flow it is on.
You could confirm that it is totally off by just removing the braided feed line from it.
Do it slowly and if it starts to drip then you know it is not off. Let it drip for a couple minutes just to be sure it is the valve that is leaking and not just pressure in the braided line.