You are thinking along the right lines, but the conclusion is not so bleak. In quantum mechanics, particles don't have positions. "Position" isn't a property of a particle at all, it is an

operator which acts on

quantum states, which allows us to determine the probabilities of experimental outcomes. The same goes for all

observable quantities in quantum mechanics, including momentum.

By projecting a quantum state onto sharply-localised

position eigenstates (i.e. by construction the position wavefunction), you can, with a small amount of mathematics, determine the probability that a position measurement will have an outcome in any given range (i.e. you can determine the probability density distribution for position measurements). Similarly, you can construct a momentum-space wavefunction by projecting the very same state onto momentum eigenstates (which look like plane waves), and determine the probability density distribution for momentum measurements.

The two equivalent ways of looking at a state - either as a sum of sharply-localised

Dirac delta functions, or as a sum of plane waves - are related mathematically by a

Fourier transform. It follows, from completely general properties of Fourier transforms, that the spread (more precisely, the

standard deviation) of the probability distribution in position space is inversely proportional to the spread of the momentum distribution, which is where Heisenberg's position-momentum uncertainty principle comes from.