The area that you mention, between the base of the penis and anus, is also where you could find the prostate.
I'm not sure how to answer the question, because I definitely have swelling of the prostate for the preiod during POIS. If it's more superficial, and sensitive to touch as you say, then it's more likely the gland. If its more internal and like a throbbing pressure that isn't too influenced by external touch, then it's more likely the prostate.
I'll answer "No" becasue mine isn't sensitive to touch.