Water filters don't soften water.

If you want a water softener, you can buy a shower stick. It's portable, and it's similar in set up to a shower filter. Super easy. It's a little pricy, but it's worth it.

Also- you should by a water testing kit: https://www.amazon.com/JNW-Direct-Pool-Test-Strips/dp/B07PFCV894/ref=sr_1_2_sspa?dchild=1&keywords=pool+testing+kit&qid=1597187623&s=lawn-garden&sr=1-2-spons&psc=1&spLa=ZW5jcnlwdGVkUXVhbGlmaWVyPUExNlpBOVhZQjJXTEg1JmVuY3J5cHRlZElkPUEwODQwOTMxM05UWDRMTU5NSVZDNSZlbmNyeXB0ZWRBZElkPUEwNTA5NjI4MzlJUUo1SklTTEEzNCZ3aWRnZXROYW1lPXNwX2F0ZiZhY3Rpb249Y2xpY2tSZWRpcmVjdCZkb05vdExvZ0NsaWNrPXRydWU=

This will tell you what factors could be irritating your skin (it checks water hardness, chlorine levels, ect.)