Thanks for using the forum. I am happy to address your questions, and my answer will be based on the information you provided here. Please make sure you recognize that this forum is for educational purposes only, and it does not substitute for a formal office visit with your doctor.
Without the ability to examine you and obtain a history, I can not tell you what the exact cause of your symptoms is. However I will try to provide you with some useful information.
A radial nerve palsy would cause weakness in wrist extension, and would not cause the sensory changes you are describing.
When one reads of your symptoms, the diagnosis of carpal tunnel syndrome, or median neuropathy, comes to mind. The median nerve can be injured anywhere along its course. If your symptoms are confined to just your hand, it may be the nerve was injured somewhere at the wrist. The carpal tunnel is the area in the wrist that the median nerve passes through. If the median nerve is compressed in the carpal tunnel, it can cause symptoms in the first three digits of the hand most often, but in some patients shooting pains can occur all the way up to the elbow. The symptoms are often worst at night. Carpal tunnel syndrome can be caused by repetitive movements at the wrist like typing, which it sounds like you do a lot of. In a minority of patients, it can be due to a systemic condition that leads to deposition of substance within the carpal tunnel (these conditions include acromegaly, amyloidosis, Gaucher's), but this is very rare and would be unlikely to present at your age. Treatment of carpal tunnel includes wearing a wrist brace, avoiding repetitive movements or optimizing the ergonomics of your key board etc. and in severe cases surgery at the wrist to relieve the nerve.
Another possibility, depending on your exact symptoms and physical examination, is a radiculopathy, though this a less likely cause at your age but still a possibility. The spinal cord is encased by bones called vertebra. Nerves start to form as they come off the spinal cord and exit through holes formed between the vertebra. If a nerve is compressed on as it exits through these holes, particularly in an area called the nerve root, a radiculopathy results. The compression could be due to arthritis of the spine or due to a herniated disc or other lesions (in someone your age, herniated discs can occur due to for example trauma). The symptoms include pain at the level of the problem (i.e. neck or back etc) and pain that may radiate down the arm or leg (depending on where the problem is). In more advanced cases, muscle weakness or sensory symptoms such as tingling or numbness may occur. A radiculopathy is often diagnosed based on history, physical examination, and MRI of the spine.
I recommend you see a regular doctor (primary doctor, family physician). After he/she examines you and obtains a history, he/she can decide whether or not you may benefit from seeing a neurologist in order to evaluate you and possibly order some testing. These may or may not include, depending on your full history and physical examination, an MRI of the neck, or an EMG/NCS, a test which tests how electricity is conducted along nerves and how muscles respond. . It is important to have this further evaluated and diagnosed, as there are specific treatments for neuropathic pain (pain due to nerve irritation) depending on the cause, and a specific diagnosis will both guide treatment and need for further monitoring.
Thank you for this opportunity to answer your questions, I hope you find the information I have provided useful, good luck.