anIsraeliYid,
one thing I heard many years ago is that one who does not shake a woman’s hand should say “I’m sorry, but for religious reasons, my wife doesn’t shake hands with men, and I don’t shake hands with women”. By phrasing it that way, you hopefully avoid insult by making it clear that the issue is not that you consider women unclean or beneath you – the restriction cuts both ways.
This is an extremely good point to think about when navigating this issue. I wonder what a good phrase would be for someone who isn’t married.