Do you or whoever else washes them, use any fabric softener. The clothes would probably feel better washed in a little unscented fabric softener. I use that on my stuff, but not always on towels, as I don't wear those, so I make the fab softener last longer by not using it on towels most of the time. I don't care for the fab softener sheets--I use the liquid version. The fab softener sheets can gunk up the dryer vent, which can cause a fire.
Another issue could be too much detergent. Try using a little less, and also try switching to a gentler one. Some of the "natural" versions are better that way, and always go for the unscented ones in detergents, fab softeners, and bleaches. I found I was bothered by the fragrances they use in those things.
Another possibility is that you may have hard water--that is, water with too much mineral content. Clothes washed in such water may pick up enough of it to bother people with very sensitive skin, who may also be sensitive to those minerals. If that's the problem, then you will need a water softener system for the whole house water supply.
Yet another possibility has occurred to me. At the risk of causing offence I will mention it, because it can be a cause of clothes not feeling good after a while. If you wear clothes repeatedly without washing them, they get dirty and then they don't feel so nice when you put them back on dirty. If you have been doing this, I suggest you wash them.
Don't let doing the laundry intimidate you. It was my mother's favorite chore. I don't really mind it, although I don't like it like she did.
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If a man does not keep pace with his companions, perhaps it is because he hears a different drummer.
Let him step to the music which he hears, however measured, or far away.--Henry David Thoreau