By the description it does sound very much like warts. As you will probably already know warts grow because of a fungal infection. As your body is having a problem fighting this off, it appears that your immune system is very weak.
I suggest that you go back to your doctor and ask them to run blood tests to check your immune system or take a biopsy to find out what these sores are or what they are caused by.
Just because the medical profession that you have been to has never seen this before, doesn't mean that they should give up on you.
Having looked at your picture it does not look like warts. But I still feel the doctor should take a swab and run some blood tests.
In the meantime, do not use any soap or shower gels or bubble baths, do not wear any creams, makeup or perfume. If you have silver jewellery do not wear this as it may contain nickel.
Only was with mild non perfumed soap and use a non-biological washing detergent.
Have you touched or been near any plant that you recall when this first appeared?
Have you been advised what to do at all with this skin problem or given any medication?
Try a little bit of natural yogurt on a small piece of your sores to see if it will help.
Have you been to a Dermatologist?