Assuming the valve is working correctly. It sounds like it is not drawing in brine. If the system was still in service for a long period of time with no salt it is most likely saturated with hardness and iron. Cloudy water with iron after regeneration is typical of oxidized iron that has settled in bottom of tank. Regeneration/backwashing stirs this sediment up and it comes in to the home. Clean the eductor assembly out on right side of valve. You can find a service manual for the Fleck 5600 on line. Make sure all the air is out of brine assembly and float. Put unit in refill position and hold the float stem down (in salt tank) until all air is purged. Get Red Out powder additive, at local hardware, dilute mixture in water and add to salt tank. Run the system every day for 1 week. This may clean the resin and restore capacity. If not you may consider rebedding the softener.
RJ

You want to clean out your brine tank as well - It may be that you are washing a bunch of garbage through your resin during the brine cycle as well.

Never mind - I see that you cleaned it out. That's what I get for skimming, rather than reading carefully.