The influence of longitudinal electron heat conductivity and finite heat exchange between electrons and ions on the drift instability in a strongly collisional plasma with finite pressure is investigated. Drift oscillations are considered whose phase velocities in the magnetic-field direction are of the order of the thermal velocity of the ions. The boundaries of the drift instability regions and the instability growth rates are determined. The dissipative effects considered result in broader regions of instability than those found when these effects are not taken into account.