What can be done about a vanity top that doesn't sit tight to cabinet base?
In my experience the normal, common way to attach such a top also fills any gaps, since it's good old silicone caulking - White or brown would seem like a good choice here. You can also attach by other means if you like and then caulk to seal and conceal. And you certainly should shim (or sand/grind the high spots on the base) to get it level. If the gap is larger than it seems in the pictures, use backer rod and caulking.