You can solve the whole thing by coughing into your hand just before you stick it out to shake someone's hand. I'm just being mean!
Here's another situation I used to have to deal with. I owned a massage school for years. Many of the more hippy, new age massage therapists insist on hugging everyone. They also don't really like deoderant. I remember a great big 6'6" man grabbing me for a big ol' hug. Because of my height, my face wound up in his damp smelly armpit. Whew! Still recoiling from that one.