> the wave is something separate from the object but is related to it.
This is a common idea in Bohmian epistemology, but, speaking as a professional physicist, I don't think most physicists would hold this position. The wave is the object, the object is the wave. "Particle/wave duality" or whatever is just bad language for "a quantum mechanical thing"
I think the point of saying it "exists at two places at once" is that if the molecule exhibited classical behavior then it either would have gone through slit 1 or slit 2, and you'd see two peaks of probability. Instead, you see an interference experiment, so the molecule went... well, here's where language difficulty comes in.
It 'went' through both slits 1 and 2?
'It' went through both slits 1 and 2?
Its wavefunction went through both slit 1 and slit 2?
Its wavefunction went through both slit 1 and slit 2 without becoming entangled with anything else?
However, whatever it did, quantum mechanics is clear about one thing: [something] [moved] THROUGH the screen! That is, because probability current [1] is conserved as a direct result of the Schrödinger equation, it definitely did not happen that the molecule vanished on one side in a puff of smoke and WHAM appeared on the other side.
So, the molecule "went through the screen", if what you mean is "at the beginning almost all of the probability was on one side of the screen and smoothly evolved to be mostly on the other". Or, rather, at the beginning you measured the position to be definitively on one side of the screen and it smoothly evolved so that later it hit the target, so that its probability was non-negligible on the other side of the screen.
Or, strictly speaking, it went forward through the screen one more time than it backtracked... You're right that the molecule didn't have to take some classically-expected "beautiful" trajectory (straight line, parabola, whatever). In fact it did all sorts of crazy things, coherently. Saying that it was coherent is a shortcut for saying that all those crazy ways' probability amplitudes sum (so that they might interfere), rather than their probabilities directly.
"occupy two places at once" is just bad language for what things do quantum mechanically. For example, "occupy" is laden with classical expectations---if a solid metal ball occupies a space, air does not. But that's not how quantum mechanics works. You can have a weird quantum mechanical state like
"ball at x and air at y"+"ball at y and air at x".
In neither addend of the sum are ball and air at the same place (just as you'd expect classically). But if you look at the sum and decide only to look at the description of the ball and forget about the air you'd decide that the ball "occupies two places at once".
Of course rather than "air" you can put "nothing" or "vacuum" or whatever. But it still isn't the case that the ball is taking up twice its volume---it's taking up its full volume, but in two alternative ways? Again, language isn't great here.
[1] https://en.wikipedia.org/wiki/Probability_current