On top of the previous good advice, I would caution against leaving soft squishy tissue on the plant. This sort of thing has a way of taking over the whole plant, given time. My suggestion would be to do careful surgery with a clean, sharp knife and cut away the soft part so that the entire margin around the incision consists of healthy tissue. In other words, completely remove the soft tissue. You might want to dip your knife in rubbing alcohol before each cut so that you don't inadvertently infect the healthy part. Leave the plant in bright shade to recover afterwards. It will be disfigured (it already is, essentially) but it will have a greater chance of survival down the road.
My Adenium mix is about half pumice. Given a pot that's barely bigger than the roots of the plant, in hours of daily sun, I find a good watering interval this time of year to be about every 3-4 days. The key (as has already been pointed out) is to be sure there is lots of rock in the mix, so that it never stays waterlogged. If you pay close attention to how much the soil absorbs when you water, you'll have a better idea of when it is going dry.
