Right arm vein hurts by my wrist. Dr. Says it might be gout but even with meds the pain is still there and so is the swelling. What could it be?

So here's the. Thing, if gout is the presumptive diagnosis a simple tap of the region (tap means doctor taking a sample with needle) would confirm that diagnosis. In the event, it's not gout, an abcess, or infection presents Similiarly, and that should be considered, a clot could develop in a vein, less common in wrist but very painful when it occurs.
Difficult w/out exam. Differentiating wrist vein and "near the wrist vein" is easy in person. Carpal tunnel, ganglion cyst, minor arthritis, or superficial vein clot are all possible and easy to distinguish on exam.
Tendinitis. Most likely at your age this is tendinitis. Rest the wtist. A splint can help. .