All of the body's extremities are susceptible to dry skin, so the fact that it is the skin on your penis that is dry may not be relevant. However, if it is very dry and cracked, then the initial advice must be to consult with your physician. There is no need for embarrassment, as any good doctor will have a lot of experience with such issues.
I often have dry skin myself, especially around the eyes. I use cucumber cream daily, which keeps this in check, although petroleum jelly enhanced with aloe vera is also very useful. When I shower, I use a gel made by Imperial Leather, made with cotton extract and oat milk.
Showering may be part of the problem - too long in the shower or bath can be bad for the skin, as can swimming in chlorinated water. You may also be cleaning yourself with a soap which is causing an allergic reaction.
Dry skin can also be caused by a lack of vitamins, especially vitamin A. Make sure you get a good supply of vitamin A by eating carrots, eggs, peas, apricots, spinach, pumpkin, oranges, liver or through a multivitamin fruit drink.
You do not specify whether you are sexually active, but if you believe you have contracted this condition through sexual intercourse, then you should visit a sexual health clinic to get tested, and tell the other person involved, as they may need to get checked out as well.
Here's a few more tips on looking after your skin, courtesy of YouTube: