This is a good reason to use a garlic press, then. You just stick it in whole, even without peeling, and squeeze it out. It never need touch your hands. And if you use it with the peel still on it, then you just have to use the point of a knife or a fork to pry off the peel and all the excess garlic comes off with it! Then open it up and lay it open in your dishwasher so the inside is face down, and that cleans the rest.
But please, don't use garlic-in-a-tube!
To get the smell off my hands, i just rub them with a stainless steel spoon (or yuou can buy one of those stainless steel "soaps", but what's the point, it's the steel not the shape that counts) and, as someone once kindly explained, i believe on these forums, it's something to do with the ions that makes stainless steel take odors off hands.
As for the cutting board, i use those plastic ones - I still have my two Joyce Chen cutting boards from the early 70s - and i stick them in the dishwasher. No smell. I would never be able to keep cutting boards separate, i just have a bunch and go through them if i need to cut something that's not going into the same pot after cutting chicken on it, or if i need to cut something like fruit after something like garlic.