Have you had any imaging of it? If you are sea-based, it sounds like probably not, but that would be the next step.
It could be something structural, like a varicocele, which is a varicose vein in your scrotum - https://www.mayoclinic.org/diseases-conditions/varicocele/symptoms-causes/syc-20378771 This has info on a specific type of physical exam to look for it.
I don't think you should take any more antibiotics without getting any lab testing done. I agree that it doesn't sound like an STD.
When are you getting off the boat? Any time soon?
There are various causes of scrotal swelling/tenderness, including but not limited to tumor, torsion, orchitis, epididymitis, epididymal cyst/spermatocele, varicocele, hydrocele, and/or hernia. If your symptoms persist, consider scrotal ultrasound.