Do most refrigerators have a filter for the drinking water my water tastes real bad coming out, but is OK at my sink

taste the water coming out of the filter ? should come from a cold water line, not the hot water line, funky smells and tastes come from your hot water heater. and I don't want to hear the hot water freezes faster bs. hot water heaters are full of stuff you don't want in your body. So if the water tastes ok....
sounds like you need to give the entire fridge a defrosting, clean it up real good. also have baking soda ready for monthly treatments, just crack a box open and stick it int the fridge or freezer.
These ice machines never did make quality ice as they just freeze still water, witch make cloudy, less than savory ice.
If you are a big ICE lover like me. a under the cabinet flowing ice machine is the best ice, as it last the longest(almost clear) and has no ill flavor.
Jay the pool pump guy longwood

Try flushing using saturated salt water or lemon

If It were me I would change the filter in the fridge, that always worked on mine.

Do not use a steel or plastic braided line. They make the ice taste bad.

Look behind the crisper in the fridge and see if it's molded inside the water reservoir. If it is you may be able to put some bleach in it to kill the mold and then run lots of water thru it to clean it out.

You will have to trace where the foul smell comes from by tracing the water line and testing the source. Check if it is before or after the filter you get the smell, if it is before the filter, you need to check if it is on the line which comes from the source or it is from the source. Most likely cleaning or probably a replacement of the part if cleaning does not help.

I would install a filter inline on the water filter I have one under the sink and on the ice maker

This happens each time you get water or only when you've recently changed the filter?  Do you live in an area with hard water/well water?  I would try cleaning the water line out with warm water and vinegar to see if this helps.  
When is the last time the filter was changed?  

then change your filter. the new filter could be different with the old filter. check the new filter, see if there is odor removal and taste removal with it.

if not then get the filter with carbon filter, odor and taste removal.

then with a new filter, run about 5 gallons of water before using the water for drinking.

tnx 4 using fixya,

drcool