Why intentionally limit the box in which you're considering your options?
If you're right-handed, why not offer your
left hand for a handshake? Really. How hard is that to consider?
It can offer you a couple potential advantages.
First, it keeps your strong hand free and unencumbered. From a martial arts and firearms instructor perspective and experience, this can be a valuable thing.
Secondly, since approx 85% of people are right-handed, it might cause a moment's confusion on the part of the person with whom you're offering to shake hands (interference in their OODA Loop) as they consider how to respond. If they're of no ill intent, then it's just a slightly awkward moment for them. Smile and look friendly and welcoming.
If they aren't of harmless intent, though? It might momentarily not only confuse them, but also "tie up" their off-side hand and let you use it as a "lever" to help you exert unexpected force that might compromise and hinder their ability to effectively use their strong hand, by turning them off-balance. Both situations might be to your advantage if they're anything other than of harmless intent.
Besides, it's not a rare thing for some people to have their strong hand filled with something (Bible, coffee, holding a child's hand, etc) and offer you their other hand. Remember how that momentarily caught you off guard the first time it happened to you? That's what you want to use to your advantage if the time should ever come that you find yourself intercepting someone trying to slip in with ill intent. It might just buy you a moment's advantage in which to decide how to act.
Think outside the box, because you can't know if a potential threat/opponent is only going to be acting within the "box" you've pigeon-holed yourself inside.
Just some thoughts. There's arguably something to be said to have a plan that allows for you to deal with everyone you meet who may not have exactly a harmless and wonderful intention.