Each weapon swings independently, with its own mods. Off-weapon mods affect both weapons, but on-weapon mods only affect the weapon on which they're present.
I assume the on-weapon rule doesn't technically apply to Smite, since you only have one weapon and hit with a shield, and because you can't wear two shields that still abides by the rule of attacking with a weapon?
for Smite info I suggest you check the Paladin forum istead, but you are kinda right.. some things dont transfer from your weapon to smite, like ED - however the +dmg from grief does which makes it the no1 smite-weapon!