If the electrons in a metal such as copper are free to mve about,they must often find themselves headed toward the metal surface. Why don’t they keep on going and leave the metal?

They don`t have enough energy to climb the potential barrier at the surface (unless you excite them with UV or something). Thermal energy is not enough. Anyway, if enough did leave due to random thermal fluctuations giving some more energy, the metal would become positively charged which would attract them back.