While I was at a show, which was crowded with people, some pervert had rubbed up against me several times. I didn't realise what he was doing at the time because it was so crowded and I thought he was just moving around. But when I left the show, I looked at the back of my coat and saw a white-ish substance on me. At first it didn't click that the pervert had ejaculated on me, so I I had reached down to clean it up with a tissue because I thought someone just spilled something on me. And it touched my skin.

My hands are chapped and sometimes have cracks . . . so is it possible to get AIDS or any disease by just coming into contact with someone's semen like that?

Unfortunately, there is a risk that germs could have been passed to your hand from the mess. That risk isn't really high, but it's not negligible either. What you can do at this point is tell your doctor what happened and if he/she deems it appropriate, you can get a blood test to see if you might have picked anything up.

And in case you haven't laundered your jacket just yet, ask your local police dept. if there's anything you can do as far as criminal charges go (they might be able to extract DNA for identification purposes).