I recommend that you see see your primary care doctor
. In general, what you are originally describing, the red spots on the penis that started right after sex, do indeed sound like a yeast infection. Most other sexually transmitted infections take a bit longer to develop symptoms, and the time course sounds right for yeast. Furthermore, things got better with the medication that was prescribed for yeast.
However, this new rash could be something else and needs to be investigated more closely. In particular, the sexually transmitted infection that could cause symptoms like those you are describing now is syphilis. Syphilis can cause a rash that spreads over the entire body and that can involve the palms of the hands, which most rashes do not. Also, syphilis can cause white spots inside the mouth like those you described.
Therefore, I would get back in to see your primary care doctor. If they feel your symptoms are potentially consistent with syphilis, they will be able to perform testing to rule this out. They will also be able to consider other potential causes of your symptoms, such as an allergic rash or eczema
. Regardless of what testing turns up, your doctor will be able to provide you with the necessary treatment so that you can put this issue behind you.