As per your description,most possible cause of your symptoms is carpal tunnel syndrome. Nighttime symptoms are more specific of carpal tunnel syndrome.Bilateral phenomenon is common, although the dominant hand is usually affected first and more severely than the other hand.
You should try OTC Advil.
Soak your hands in warm water for 5-10 minutes twice a day.
You can try wrist splint at night.
Keep elbow straight while sleeping.
Avoid putting pressure on arm while sleeping.
Rotate your wrists and stretch your palms and fingers
You can do gentle massage with topical diclofenac sodium ointment from neck to fingers.
Avoid prolonged hours of computer,mobile use,writing,etc.
A clinical exam by Orthopedist can be advisable.
Do stretching exercises.
Physical therapy can be effective in your case.
Feel free for further queries. It's good to discuss rather than quit.
Your satisfaction is our goal.