This sound very uncomfortable! I strongly recommend that you get in to see your primary care doctor
as soon as possible. There are effective treatments for this problem, and your doctor can help!
It sounds like you are probably suffering from eczema
. This is a condition in which the skin of the hands becomes dried out excessively, leading to redness, itching and, eventually peeling and cracking of the skin.
There are several phases to treating eczema. First of all, you have to protect the skin from additional damage from drying. This means frequent moisturizing of the skin with a good, thick moisturizer. It also means avoiding washing your hands more frequently then you need to, and avoiding strong chemicals and soaps.
In addition to this, however, most cases of eczema that are as severe as your require medications to clear up entirely. In particular, a topical steroid cream or ointment could bring significant relief of the burning and itching and also help clear up the inflammation rapidly.
Start by seeing your primary care doctor right away. They can confirm the diagnosis of eczema and also prescribe any medications that you might need.
Good luck, and I hope you feel better soon!