In all probability the swollen or sensitive cord like thing you can feel is epididymitis or inflammation/infection of epididymis and since you do not have any pain and redness in scrotum, it is most likely to be a chronic infection. It is usually due to bacterial infection, but if you have been sexually active then it can also be due to Chlamydia.
Another possibility is beginning of varicocele or enlarged veins. Do discuss this with your doctor and get this evaluated. Treatment will depend on diagnosis. And, it is unlikely to be cancer, but yes, needs treatment. Take care!
The medical advice given should not be considered a substitute for medical care provided by a doctor who can examine you. The advice may not be completely correct for you as the doctor cannot examine you and does not know your complete medical history. Hence this reply to your post should only be considered as a guiding line and you must consult your doctor at the earliest for your medical problem.
I have a doctor's appointment tomorrow. I am a virgin, although I do masturbate. The doctor's appointment is for what I think are really bad allergies, but I'm hoping I can mention this problem too. I really appreciate your advice!