Filter water in a tank without a whole house filter
It sounds like you have really hard water. https://en.m.wikipedia.org/wiki/Hard_water

This means that the water has a lot of dissolved minerals in it. These minerals cannot be removed with a filter and actually tend to clog filters and shower heads. You can buy certain soaps and detergents that claim to work better with hard water but their effect is not as good as a water softener.

Except for softening the water your options are limited to a different soap or other source of water. You might look for a source of fresh water close by such as a lake or river that has softer water for bathing and hair washing.

Ironically, people love mineral water to drink but it is not very good for washing.

Good luck!