I just want to mention the possibility of more advanced problems just in case, since you are having problems getting treated.
Hypersensitivity - hard to tell if it's just the skin (as in skin irritation) or the nerve underlying innervation of the penis.
If it's the nerve (and not just the very external layer of skin) then it may be pudendal nerve problems. This is the nerve that runs from the spine through the alcock's canal near your bottom, through to the exterior portions of your ****, testes, and penis to innervate them.
Problems along this nerve line can cause pain, numbness, or hypersensitivity along this nerve route.
Problems can include lower back problems (anywhere from L4 through S4) tugging on it, ligaments rubbing it the way, alcock canal getting smaller and rubbing it (scar tissue or by other means), piriformis muscle problems since it runs close to it, etc. It can even be from Femoroacetabular Impingement (FAI) or hip impingement syndrome. So if you have any other symptoms like these check them.
As a female, I have pudendal nerve entrapment problems from L5 S1 disc degeneration. So, yeah, anything along that route can upset it. Sometimes my clitoris (on you it is would be the same with your penis) would get hypersensitive in not so nice a way. I didn't even want to touch it or have anything like underwear rub it. On other places along the nerve line, I swear it felt like the skin was highly irritated, but the skin was actually fine. It was the nerve that was having problems. Figured out it was my spine pulling on it from above by being a bit out of place through an MRI.
So, if your symptoms progress to other things, too, thought you may want this info.