Drying out and cracking of the skin of the hands during the winter is a very common medical problem. It is very unlikely that it is related to medications or hormones. Instead, the dry air and cold strips away the natural layer of protective oils that normally cover the skin, leading to drying out of the skin and cracking.
In order to minimize this, you should make sure to liberally apply a skin moisturizer multiple times a day. Bag Balm and other "sticky" moisturizers are better because they are thick and tend to trap in the skin's moisture a little better.
You should also avoid things that worsen the drying process. By far the most common culprit is water. You should always where rubber gloves when washing dishes, and you should limit how often you wash your hands, never use very hot water, and always pat your hands dry instead of rubbing them dry. If you are doing yard work or house work, a pair of light cotton gloves can keep your hands protected.
If there are any areas of redness, scaling, and itching, you can apply an over the counter steroid cream to these areas to clear them up.
If none of this works, see your primary care doctor
or your dermatologist