Previous models of solid grain melting in solar nebula shocks have neglected gas cooling behind the shock front; i.e., they considered adiabatic shocks. The effect of gas cooling in the postshock region is studied in this article. It was found that shocked gas is cooled efficiently by dipole molecules and small dust particles, and this results in a sharp increase in gas density in the postshock region. Submillimeter and larger grains cross the region of cooling before being decelerated, and are heated by the drag (through the cooled and compressed postshock gas) more strongly than in an adiabatic shock. This effect opens the possibility of melting of millimeter-size dust aggregates (chondrule precursors) in that cooler region, even when the matter is transparent for the thermal radiation of the aggregates. The possibility of formation of chondrule precursors in the collapsing presolar cloud and of melting of the precursors in the radiative accretional shock during formation of the solar nebula is speculated. In such a shock, grains could be melted at heliocentric radii up to the inner part of asteroidal belt if the density of infalling gas were only few times larger than the average density of infalling gas corresponding to an accretion rate of 10-5M⊙ year-1; the limiting radius for melting increases with density. Clumpy accretion and enhancement of density in the vicinity of the centrifugal radius of infalling gas provide the necessary densities of infalling gas. Enhancement of drag heating due to fast cooling of postshock gas is also significant for shocks of other origin in the solar nebula. At the central plane of a minimum-mass solar nebula, chondrule precursors could be melted in shocks with velocity ≥10 km sec-1. The duration of the molten state and the cooling rate of a grain are determined by the grain's kinetic energy and the rate of its decay (not the time scale of radiational cooling for the grain!). Millimeter-size compact grams have temperatures ≥1600 K for <103 sec, and ≥700 K for ∼104 sec, times that are appropriate for chondrules. These time scales could be larger in faster shocks when grains have larger kinetic energy, Chondrule precursors could be formed as fluffy aggregates before and during the collapse of presolar cloud. Later they must increase their density. The possibility of increasing the density of the aggregates by means of mutual collisions or as a result of surface tension when they began to melt is discussed possibility of increasing the density of the aggregates by means of mutual collisions or as a result of surface tension when they began to melt is discussed.