The idea of a quantum field as a bunch of oscillators is really quite bad an analogy if you want to use it to reason about how quantum field theory actually works, see this question and this question. However, this is quite irrelevant in the context of this question because this question isn't actually about how quantum field theory works, it's just about how quantum mechanics works - that we can model a free particle by a wavefunction that obeys a similar wave equation to classical waves is after all the "wave mechanics" approach to elementary quantum mechanics, not something particular to QFT.
The question seems to think "the position" of a particle to be associated with a relatively narrow "spike", and that particles should always have this kind of wavefunction because after all we observe particles to be localized. This is not at all how quantum mechanics works - in principle the wavefunction of a particle can be arbitrarily spread out, and only when we measure position we get some sharply localized result and force the wavefunction of the particle to start again from being a spike at the location where we measured it. There does not need to be an identifiable "position" of the particle prior to measurement, not even approximately, and there is no motion in the traditional sense involved with this action.
If your instinct is to ask "how" the measurement does this, exactly, then that's the realm of the measurement problem, to which standard quantum mechanics just gives no answer at all - modeling "how" this happens is not required to describe what happens. The wavefunction simply gives the probability amplitude to find a particle in a particular location if we look. If we don't look (where "look" might mean any sort of interaction with the particle that allows inference about its position), we can't assign a definite position to it and the wavefunction may remain arbitrarily spread out.
And indeed, when you take a sharply localized spiky wavefunction for a free particle and just don't measure position for a while, the wavefunction of that particle will spread out exactly like a classical wave - computing this is a popular exercise in elementary quantum mechanics textbooks.
