The question of the soul has really nothing to do with it.
I ask again: If you get killed in a chamber, would you expect to wake up in another chamber, simply because that other chamber constructed an exact clone of you? No matter whether you believe that souls exist or they don't, the answer is no.
I would expect it, if it's happened to me before!

If I have the memory of going into a chamber and exiting in another part of the world, then I would expect it to happen again.
If memories are stored as electric impulses in the brain's synapses, wouldn't the presumedly electron-beaming of the body's molecules, including synapses, necessarily reshuffle all the data of every single stored memory in your brain? The teleportee would almost certainly emerge from the arrival end of the process as a brain-dead jibbering idiot, no?
Well, the memories are stored in synapses. These are neurons that are separated from each other, and they have different ions available to modulate their interaction. If the cells/ions/etc. were all transported, then you'd still have the same mind as before. Or, at the very least, similar enough that it would not be confusing. I mean, after a vigorous sneeze, you change a lot of those synaptic relationship in subtle ways. Personhood is retained however
Your fuzzy memory is spot on. The "no-cloning" theorem of quantum mechanics postulates, that a full copy of a state is impossible. So any transfer retaining all information would have to destroy part of the information in the old body. You could* transfer all information to a new body by quantum teleportation, but you would never end up with two exact copies.
But that is only a limitation if an essential part of our consciousness is of a quantum nature. If one could make a close enough copy with classical information alone, the copies might be the same self, even if they're not exact copies. At the moment there is not information to decide either way.
*"You could" in this case means "it does not violate a fundamental theorem of modern physics". At the moment, however, a theorist would not be able to figure out how to do it, or even if this is possible. And an experimentalist would declare you mad if you suggested trying it.

You mirror what I've been saying. One could functionally
copy a mind such that the mind was capable of cognition. If this resulted in the destruction of the original person, then there would be (effectively) a death followed by a new birth. The clone might not experience a break in consciousness, but there would be one.
However, if the copying process was so fine-grained that the destruction of the original was physically required (in a QM sense), and the process of interpreting the copied information also required the destruction of the storage media, then I think it's fair to say that the person has been
truly teleported. As much so as if they'd just walked across the room.
QM, I'm pretty sure, is too fine-grained for simple consciousness. Unless I am misunderstanding. Transcranial magnetic stimulation, otherwise, would be killing people without their knowledge.