Both are true. Copying a quantum state means moving it from one object to another, which turns the target into the source, and the source into... something else. If we managed to do that at a full body scale, a "you" would appear at a target location, while a bunch of "something else" would be left at the source location.

An external observer would say "you moved", turning a pile of target "something else" into you, and leaving a pile of "something else" at the source. You yourself... well, as long as you don't worry too much, you would also perceive having just moved from source to target.

Still, there remains that pile of "something else" that used to be you at the source location... but as long as everyone, including you, don't decide to call it "a corpse" or "your previous you's remains", everything would be fine.