I'm an old fashioned lady, even though I carry a gun.
A gentleman should NEVER extend his hand to a lady without invitation. If a lady offers her hand to a gentleman, he may then take hold gently - but it is poor form to actually "shake." Kissing the hand is no longer encouraged much, but it is a sure way to charm an old lady most of the time. Just don't initiate it. You gentlemen have to figure out your own criteria, but handshakes with strangers for no real reason strikes me as strange.
It is important, I think, to reserve "handshakes" or other such intimate behaviors for serious purposes, not something without real meaning or consequence.
This germ thing has gone far beyond silly. You live in a world awash with germs of all kinds, on absolutely every surface not found in a surgical suite... and all too often even there. You only need to remember to keep your fingers out of your mouth unless you've just washed YOUR hands. Unless you wear surgical gloves, your hands are contaminated by... and contribute to the contamination, of everything you touch. Washing your hands after known contamination is important, but it only cuts down on KNOWN contamination and doesn't last very long.
The alcohol and "gels" do not actually kill bacteria and wouldn't even make most viruses sick.
If you use the alcohol and gels much, you are breaking down the natural skin oils and friendly bacteria which allows pathogenic bacteria and viruses an open door to contaminate a deeper layer of tissue. You can't wash that off without a great deal of effort and time.
I spent part of my nursing career in infection control, and you don't even want to think about how many "germs" wash in, over, under and around your body everywhere, all the time. Learn how to maintain a solid, healthy immune system. Then very few of the "germs" can bother you, no matter how you come in contact with them.