English Speaking Dermatologist?
I think that most people involved in a medical field have enough English for you to be understood. Alot of the terminology is English and they have to know the English in order to qualify. 피부 (pee boo) clinics are on every other block. Of course you need to get past the receptionists, but just pointing at your problem I've found does the trick!
If you luck out just try another one around the corner.
(btw if you enter 'dermatologist' in the search bar you will see a ton of hits for the same question and plenty of recommendations).
I know a very good skin Doctor who speaks English well , He is located in Dushil . That is on the no 1 subway line stop no 131 , That is 3 stops after Pnu . I am not sure what exit it is . But its opposite the Islam center , Its in the building of Busan Bank , The clinic is on the 3 rd floor , Its Dr Kims International Skin Clinic , If I find out what exit it is I will post it in a week or so , He deals with all skin and hair issues ,
I went to good moonhwa hospital just near Beomil Station for something similar. A lady called Ella works there and her number is 010 9345 4556. She speaks good English but the place is like a lot of hospitals here and more of a business than doing what's best for you. They seemed a bit expensive and I didn't go ahead with the treatment. You could also just go to the Busan Medical Information Center in Seomyeon. They can give you a few places to visit. Contact me if you want their details.
As I mentioned Dr Kims International Skin clinic
Located Dushil 131 that is no 1 line
go to exit 5 or 7 its on the 3rd floor of the pusan bank bld
Dr Kim's hours have been changed alittle , On Saturdays they close at 3 pm , All other days they close at 6 pm , Please note on Thursday they dont open in morning ,