We observed 51 sources in the Q-U-I JOint TEnerife (QUIJOTE) cosmological fields that were brighter than 1 Jy at 30 GHz in the Planck Point Source Catalogue (version 1), with the Very Large Array at 28-40 GHz, in order to characterize their high-radio-frequency variability and polarization properties. We find a roughly lognormal distribution of polarization fractions with a median of 2 per cent, in agreement with previous studies, and a median rotation measure (RM) of ≍1110 rad m-2 with one outlier up to ≍64 000 rad m-2, which is among the highest RMs measured in quasar cores. We find hints of a correlation between the total intensity flux density and median polarization fraction. We find 59 per cent of sources are variable in total intensity, and 100 per cent in polarization at 3σ level, with no apparent correlation between total intensity variability and polarization variability. This indicates that it will be difficult to model these sources without simultaneous polarimetric monitoring observations and they will need to be masked for cosmological analysis.