Hard to believe that your salinity is increasing that much just from fill water. AT the numbers you are citing, you would need to add 9lbs of salt per month to see the increase you have. That just not realistic.
Can you test the salinity of your raw water (before the softener) and the salinity of the water out of your softener using your K-1766? You may need to use a large water sample, like 40mL, to get a 50ppm/drop precision. Small glass beaker, 50mL size, work well for this. All you need to do is add 3-4 drops of the indicator reagent and then just count off the titrating reagent as you normally would. With a properly operating water softener, you should really see no difference between the input and output water salinity since the test is only sensitive to chloride ions and not sodium ions.
If you are seeing a big difference in salinity across the softener output, then your water softener may be defective. The high salinity brine solution used to regenerate a water softener is only used during the regeneration process and it should be completely flushed out with a backwash and rinse step after the media is soaked in brine. There can always be little bit of residual brine left over, but it should quickly clear out after regeneration.