Okay, you are pretty young to be having chronic pains that arent a direct cause of an injury. For starters I would urge extreme caution when dealing with and blindly following your traditionally educated doctor. Dont get me wrong, they have their place but I dont believe a regular doc is going to do anything but offer you life long permanent damage.
Next, lets look at inflammation, what is it? Inflamation is one of the steps in the bodies natural healing cascade, its necessary for proper healing!! As a matter of fact its an indicator of healing. Sure at times too much inflammation can be detrimental such as in blunt trauma etc but for the most part chronic inflammation is telling you there is damage thats in the process of being healed.
I assume you have been dealing with this issue for some time so then one has to ask, why is it not healing?
Tendons. ligaments, skin, arteries, meniscus and various joint linings and connective tissue are all collagen based tissues. Generally speaking these tissues are not very vascular and have a slow cellular turnover rate (ie it typically takes 300 to 500 days for all the cells in a tendon to be replaced by the body under normal circumstances). When one of these tissues is injured additional immune cells and other factors pool at the site of injury to facilitate repair. The less vascular the tissue the slower the healing.
R.I.C.E commonly preached does what? rest, ice, elevation and drugs all do what? They reduce circulation! Why do you want to reduce circulation to an injury thats already not very vascular and has limited blood flow? Generally pain comes with inflammation so reducing inflammation reduces pain BUT it also reduces healing!!!! This is fine for the typical couch potato but not for an athlete!
Avoid ice, nsaids/anti-inflamatories, they will make it worse.
Stop activities that aggravate it and make you want/need the above ice and nsaids!
If you have done the above the next step is to make sure you body has the building blocks of the damaged tissue. Primarily vitamin C in the form of ascorbic acid(caps or powder is fine) multiple doses throughout the day, generally 1-3 GRAMS per serving, lysine 5-6g per day, proline 2-3g per day and chondroiten 1-2g per day. The reason for the high dose vit C is because its required for collagen synthesis but its also an electron donor to the immune system and detoxifier. Some people are exposed to so many toxins daily taht even if they take >20grams of vitamin C per day it can all get used up by the immune system and you still may be lacking whats needed for proper collagen synthesis.
Next, id recommend trying to find a good chiropractor who also deals with massage and physical therapy. In addition to nutritional shortcomings you probably also have a shoulder imbalance and mobility issues which are causing chronic pain because the shoulder is not functioning properly. extensive rotator cuff exercises combined with mobility work and fascia work to break up any adhesion limiting movement will go a long way to full shoulder recovery.
I know it sounds like a lot but your young and if you ignore it and continue to work through the pain I guarantee 10 years from now you will have permanent and irreversible damage. Dont ignore it.
Your wrist issues since so short lived will probably be fixed by a little rest and the supplements I listed above but it may take some time. They key is to avoid stuff that aggravates them.