@ddljohn-- I'm not a doctor and I think it's time for you to see a dermatologist about this if you haven't done so yet.
I'm inclined to think that you either have a skin yeast or bacterial infection causing the rash, or you are experiencing an allergic reaction to sweat. None of these are rare but you need to see a doctor for the correct diagnosis and treatment.
Do you get the rash anywhere aside from your armpits? Such as in the groin area? If you do, this is a sign of a sweat rash (miliaria). This often happens in hot weather when someone remains with sweat on their skin for a long time. The treatment is taking frequent showers and using corticosteroid ointments as necessary.
If you don't have the rash anywhere else, but if it is persistent despite changes in body wash, soap and deodorant, it could be a yeast infection or a bacterial infection. If so, you will need medicated ointments such as anti-fungal creams or antibacterial creams.
There are many possible causes of an armpit rash. A deodorant rash is common but other rashes can look very similar. So see a dermatologist and let the doctor diagnose the rash.