Performers should sign autographs, period. In a career as volatile as Broadway, they should appreciate fans and be gracious for 15 minutes after the show. It's not a lot to ask. Unless they're rushing out for a good reason, they shouldn't be so uppity. I have a lot of respect for performers who spend time taking photos and indulging their fans, and little respect for those who act like divas offstage.
And you don't get ill from handshakes. Unless someone transfers their sputum to your hand then you ingest it. Which is just stupid and gross because you shouldn't really lick/eat your hands.
"Jane, I've been dealt a blow - I've been dealt a blow, Jane."