This sounds a lot like testicular torsion. It's happened to me and it can be painful enough to make you pass out. When I have experienced it I got pains shooting up into my stomach and my testicles were so sensitive that they were in constant, agonizing pain.
If you're having a torsion then you need to see a doctor immediately to get it corrected or you risk losing the testicle. My doctor told me that if it goes for more than 12 hours then the decreased blood flow can cause atrophy and in some cases can essentially cause the testicle to die, I'm sure there's a better term for what I'm referring to.
I don't recommend it, but after it happened to me a few times I figured out a trick that seemed to fix it for me. It was still very painful and it probably would have been smarter to go to the doctor to get it corrected. I found if I laid down on my back while not wearing any pants or underwear I could lift the loose skin of my scrotum up to the point where it would also lift my testicles up. When I released it they tended to fall back into place not twisted anymore. It usually would help if I had my knees bent or lifted my legs off the ground. I have a very loose scrotum so I don't know if that made this fix more possible, but my urologist told me that it was probably part of the reason that I was susceptible to torsion in the first place.
There is a procedure that can be done to prevent torsion, but it sounds very unpleasant and I opted not to get that despite my urologists recommendation. Essentially, the suture the testicles together so that they are unable to be twisted. The idea of having what felt like a single testicle that was the size of two testicles was pretty off putting to me, not to mention the recovery from the surgery.
I found that it happened to me more often if I was wearing tight-ish pants, and especially if I was doing some sort of work that required me to bend over at the waist a lot. It sounds counter intuitive, but I recommend avoiding wearing anything that hugs your nuts too much.
Go see a different doctor. If your doctor doesn't recognize the symptoms of a torsion then you need to see someone else who can diagnose it. Don't downplay the pain you're experiencing, its supposedly one of the most painful things you can experience aside from breaking your femur or getting impaled.
