Warning: I'm a prrogrammer, not a physicist.
IIUC, one interpretation is that "yes, it's a particle, but it only has a probable location/velocity/momentum/etc.". So it is simultaneously both a particle (as an object) and a distribution of probabilities as characteristics of the particle. It's the probabilities that move as a wave, but it's only the particle that we can detect.
N.B.: I studied this quite a long time ago, so not only are things a bit fuzzy, the "best" way of looking at them may have changed.
Also: Please note that this is just an interpretation of the data. The data appear to be such that multiple (wierd) interprations are possible, and no-"non-wierd" interpretation is possible. My favored interpretation is a variation of the Multi-World (EGW) interpretation modified to include multiple pasts as well a multiple futures so the the universe becomes a directed graph with a (possibly unique) origin and a (unknown) limit. But most state transitions though probabilistic don't make the universe larger because the multiple pasts of each instant-instance merge an (essentially) equally large number of world-lines to the divergence towards the future. So the number of world lines stays approximately constant. Think of it as a really huge state transition table with probabilistic transitions being processed on a system with a truly huge number of independent threads. And all exits from each state are taken with a weight equal to the probability. This is already pretty messy, but then you need to start worrying about the light cones, and the fact that information transmitted via light only experiences time when and absorbtion/re-emission even occurs. (I haven't yet figured out how to handle light slowing down when not traversing a vacuum but also not being absorbed. Does it start experiencing time?)