We use Milky Way-like chemodynamical simulations with a new treatment for dust destruction and growth to investigate how these two processes affect the properties of the interstellar medium in galaxies. We focus on the role of two specific parameters: fdes (a new parameter that determines the fraction of dust destroyed in a single gas particle surrounding supernova) and Cs (the probability that a metal atom or ion sticks to the dust grain after colliding, I.e., the sticking coefficient) in regulating the amount and distribution of dust, cold gas and metals in galaxies. We find that simulated galaxies with low fdes and/or high Cs values produce not only more dust, but they also have a shallower correlation between dust surface density and total gas surface density, and a steeper correlation between dust-to-gas ratio and metallicity. Only for values of fdes between 0.01 and 0.02, and of Cs between 0.5 and 1 our simulations produce an average slope of the dust-to-gas ratio versus metallicity relation consistent with observations. fdes values correspond to a range of a total fraction of dust destroyed by a single supernova between 0.42 and 0.44. Lastly, we compare predictions of several simulations (with different star formation recipes, gas fractions, central metallicities, and metallicity gradients) to the spatially resolved M101 galaxy, and conclude that metallicity is the primary driver of the spatial distribution of dust, while dust-to-gas ratio controls the cold gas distribution, as it regulates the atomic-to-molecular hydrogen conversion rate.