I think you need a correct diagnosis so that you can be treated effetively. It sounds like you'll need to contact him again and politely report that the cool compresses have not been effective and ask for a diagnosis (what he thinks are causing these symptoms).
From what you say in your post you have eczema so perhaps this is related. Maybe you could find a dermatologist who specializes in eczema or related skin conditions but this should not be necessary.
It's disappointing how many doctors seem unable to help with diseases or conditions well within their specialty. Then when all seems hopeless, you suddenly find a good doctor who has the answer.