Sounds like soap residue or a lot of total dissolved solids in your water. Have your water tested both before and after the water softner. The softner doesn't put salt in the water but it can pick up a slight amount of salt residue from the resin cleaning process but usually not enough to notice or affect your health.
The softer water should make cleaning the residue easier so I suspect the softner isn't cycling properly or soap residue second. The softner should take out some of the TDS but maybe not all. Start with the test.
Also some homes are plumbed with only the cold to the kitchen sink without going to the softner (for taste reasons). Usually they will plumb the hot water so all lines are softened. Check the plumbing. Sometimes a single handle faucet (kitchen and bath) will go bad and allow water to bypass such as hot to cold or cold to hot. That is usually more noticable when it happens at the shower. Close the stop on the hot side of the kitchen faucet and then see if you get cold to come out where the hot is supposed to.
Is this a new problem that didn't happen before or has it always been this way? If this is a new and recent problem, then most likely it is the softner isn't regenerating right. Sometimes the nozzle to the brine line will plug up and not get any brine to clean the resin everytime it regenerates. Assuming you have the type with 2 lines to the brine well, during the brining process, one line will flow water down the brine well where a nozzle causes a venturi effect which sucks up brine that returns to the main valve for cleaning the resin. There is a way to check it but I'd want more info back before I get into it.