Regular people can still find trash and detritus from civilizations hundreds or thousands of years old. So yes, the evidence could still be there.
As for erasing Earth, if they're all 100% in agreement with this, then it could be taken care of in the first generation, just teach all the children, "This is how it always has been and always will be." Any comments about coming from another planet will probably be written off as dreaming/insanity/senility, depending on the person saying it.
The biggest problem with crashing the ship into the ocean is that 1) it would have to be done somewhere well away from their settlement, to avoid damage from tidal waves*
, and 2) it sounds like a suicide mission. Would it work better for the story to have the ship crashed into a lake closer by? Either way, a ship designed to travel star systems will probably be robust enough that evidence, if not the entire ship, is still present a thousand years later. It also begs the question of what happened to all the shuttles used to ferry them down to the surface? If the colonists really
wanted it destroyed, they should have pointed it at the sun and let it drift on down.