The effects of scalar and pseudoscalar ultralight bosonic dark matter (UBDM) were searched for by comparing the frequency of a quartz oscillator to that of a hyperfine-structure transition in ^{87}Rb, and an electronic transition in ^{164}Dy. We constrain linear interactions between a scalar UBDM field and standard-model (SM) fields for an underlying UBDM particle mass in the range 1×10^{-17}-8.3×10^{-13} eV and quadratic interactions between a pseudoscalar UBDM field and SM fields in the range 5×10^{-18}-4.1×10^{-13} eV. Within regions of the respective ranges, our constraints on linear interactions significantly improve on results from previous, direct searches for oscillations in atomic parameters, while constraints on quadratic interactions surpass limits imposed by such direct searches as well as by astrophysical observations.