First, I agree with Salcott to find a good teacher, if you don't already have one.
Second, don't assume you have anything resembling or will develop CTS. You could simply have a case of tendinitis. Regardless, get it checked out. If it is CTS and you catch it early enough, it's not that bad an issue to correct, given you take the correct course of action. If it's tendinitis, you may need to figure out how to reduce tension in your arm.
This sounds like, as you already said, a lifestyle issue. You may need to set your priorities on this. If you are really serious about bass, you may need to consider quitting throwing shot put. I know it's difficult; I quit wrestling back in high school because of health issues related to the bass. I no longer play Ultimate Frisbee as much, because of the strain on my wrists.
Try massaging your arm near the elbow, especially on the side closest to your hand. If this is painful, you may have tendinitis...it often stems from back here. Another possibility is cubital tunnel syndrome, but I don't know how shot put would cause that. Anyway, get it checked out (even an urgent care would suffice...they would tell you if it's serious enough to warrant a physical therapist or other tests), investigate wrist braces, and figure out your priorities. Whichever you choose (if you have to, and I hope you don't), you will have to make decisions based around it.
Hope this helps.
Peace,
RJ
