What's embarrassing about using 1 yen coins?
I'd rather make use of those small coins than walk around with my pockets full of shrapnell. Naturally, the ideal is when the available coins allow one to pay with the exact change... but failing that, I just aim to end up with fewer coins in my pocket than I started with.
Sometimes the store clerks get confused with the somewhat weird amounts that I hand over when making a purchase. It doesn't embarrass me. Once they punch the number into the till and realize that they're going to give me a single 500yen coin in change, they understand my game.
As for what to do with the coins that are left when I leave the country. Isn't that what the charity boxes in airport terminals are for?