Hi I like the soaps from the natural health food store such as the olive oil soap or pine tar soap. It just has to be kept dry after use to avoid gumming. There are other really nice all natural ingredient soaps such as lavender soap or rose oil soaps. The natural ones seem less likely to leave a rash. Also the it can be used in the sweat areas and private areas and left mostly off of other places if it is really bad. Sometimes there is a chemical in the water itself that can cause rashes. A water distiller might be useful for this problem. Large ones can be obtained if needed.
If enough water is drunk then there is less likely to be a problem with an inner dry nose.
Sometimes too much moisture on the skin or shaving substances can cause rashes. The skin needs to be allowed to dry out after bathing particularly.
One needs to be sure it is really the soap that is the problem so if after changing soap there is still a difficulty perhaps a dermatologist or even eye ear nose and throat doctor can be consulted.