We present a model atmosphere analysis of cool hydrogen-rich white dwarfs observed at the Very Large Telescope (VLT) with the X-shooter spectrograph. The intermediate-dispersion and high signal-to-noise ratio of the spectra allowed us to conduct a detailed analysis of hydrogen and heavy element line profiles. In particular, we tested various prescriptions for hydrogen Balmer line broadening parameters and determined the effective temperature and surface gravity of each star. The spectra of three objects (NLTT 1675, 6390, and 11393) show heavy elements (Mg, Al, Ca, or Fe). Our abundance analysis revealed a relatively high iron-to-calcium ratio in NLTT 1675 and NLTT 6390. We also present an analysis of spectropolarimetric data obtained at the VLT using the focal reducer and low dispersion spectrograph (FORS). We establish strict upper limits to the magnetic field strengths in three of the DAZ white dwarfs and determine the longitudinal magnetic field strength in the DAZ NLTT 10480. The class of DAZ white dwarfs comprises objects that are possibly accreting material from their immediate circumstellar environment and the present study helps to establish the class properties.