The slick feeling from soft water is just your own natural skin oils. Hard water makes soap scum on your skin with the soap and strips more of those oils away, thus the difference in feel. Took me a few weeks to get used to the difference in our current house, now I prefer it, though I know plenty of people that just can't stand it. I get much less itchy when the air is cold and dry out when I shower with soft water vs hard.
If you look into "salt-free" softeners, just do your research and keep in mind they're not actually water softeners, they're water de-scalers. Unlike softened water, the hardness is still present in the water it is just less likely to adhere to surfaces. You miss out on some of the benefits of soft water, I but don't have the increased sodium (or potassium) or slick skin feeling, and like softened water, won't have the mineral build up on fixtures you'd get from not using anything.
If your reason for not wanting to soften with salt or potassium is taste, or having increased sodium or potassium in the water, you can always put a reverse osmosis system in your kitchen for drinking water which will strip those out. That's the route I went, but it does mean yet one more device in you water system. Hardness aside, if your water just doesn't taste good, RO filters are great for that as well.
I built my own water softener from components (Fleck valve), and have been happy with it.