Finding Skin-Friendly Handwashing Solutions in Public Places

I am curious about how others manage handwashing in public places. Personally, I've found that the soap in most public washrooms is very harsh and often contains fragrances that irritate my skin. I always carry a small bottle of hand sanitizer in my bag, but it can sting on open wounds and often dries out my skin even more. Any suggestions would be greatly appreciated.

Thanks!