As quantum physics says the particle behaves differently depending upon when we look and we don't. When we look, it behaves like a particle and when we don't, like a wave. And uncertainity principle states we cannot simply determine the position of the particle due to its wavy nature, but at the time when we calculate its position it already behaves like a particle rather being like a wave, so why can't we determine its position at that instant ?
We can determine the position of a particle exactly but that is only when we measure it. When we measure the position of a particle, the wavefunction collapses and we find the particle at a particular definite location. But, what we mean when we say that there is an uncertainty in this measurement is not that we are unsure as to whether we really found the particle at the position we found when we made the measurement but that if we had a large number of identical wavefunctions and we had made measurements on each of them then all the experiments would not have found the particle at the same position. There is a variance in this data of positions that we find when a large number of identical wavefunctions collapse. This means that at an instant when we are not making a measurement, we can't say that the particle is at a particular position because the outcome of measuring the position could give me a number of results of which I can only know the average and variance but not the exact outcome. Indeed when we actually make the measurement we will get a particular value and that is the definite position we make the particle to take through the act of measurement. This whole story seems completely consistent and coherent with the picture of the wave and particle nature that you have in mind.