Thank you for taking the time to write is and ask about this condition. I think you would be surprised to know that many people struggle with this or similar reactions.
While therapy and medicine are certainly valuable approaches to dealing with this anxiety-based reaction, I believe there may be a more direct method to try initially. It this doesn’t work you may want to begin therapy or get a medical evaluation. But the first place I would begin is with biofeedback.
Biofeedback (short for biological feedback) uses a machine to give you visual or auditory feedback from areas of your body that are typically not under our conscious control. The list of conditions that biofeedback has been used for is extensive, not the least of which is shaking, sweating, blood pressure, heart rate, and many others.
The concept is simple enough in that by using instruments that note slight changes in subtle features of the skin or muscles (as an example) you can consciously train yourself to reduce anxiety and manage your condition. I would recommend working with a person certified in biofeedback. Here is a link to their association. I would encourage you to find someone in your area and try this treatment. Typically the learning curve is fairly brief, and you may be able to get some direct control over your general anxiety, and your hands.
My sincerest hope is that you will be able to conquer this anxiety and go on to pursue the profession of your dreams.
Wishing you patience and peace,