This can be carpal tunnel syndrome. Nighttime symptoms are more specific of carpal tunnel syndrome, especially if your symptoms are relieved shaking the hand/wrist. Bilateral phenomenon is common, although the dominant hand is usually affected first and more severely than the other hand. Read the details here;
The use of ultrasound, phonophoresis, iontophoresis may provide relief. It may be possible to enlarge the carpal tunnel by specific stretching techniques. A specific stretching/strengthening program for the hand and wrist may be useful in improving strength. Patients whose condition does not improve following conservative treatment should be considered for surgery. Surgical release of the transverse ligament free the median nerve and improve the symptoms.
Another possibility may be pinched nerves in neck. Read here;
Cervical traction, deep heat therapy, physical therapy and anti-inflammatory analgesics are some of the part of treatment protocol.