12CO (1-0) observations of two Southern dark clouds (globules) associated with cometary nebulae GRV 8 (a biconical nebula) and GRV 16 (a conelike nebula) are presented. GRV 8 shows an outflow from the central part of the nebula (where in 2MASS images a star is located, which is perhaps responsible for this outflow); however, both lobes of the outflow are redshifted with a velocity of +1.95 km/s with respect to the molecular cloud. The two opposite redshifted lobes are a rather rare phenomenon that could be explained by the presence of a double star instead of a single one as the engine responsible for the outflow. The two lobes are almost parallel to the axis of symmetry of the biconical nebula. In the case of the conelike nebula GRV 16 we observe a bipolar outflow, where the eastern blueshifted lobe has a velocity of −4 km/s with respect to the molecular cloud, and the western redshifted one has a velocity +2.5 km/s. The outflow has a direction almost coinciding with the axis of symmetry of the conelike nebula. The star associated with the conelike nebula is responsible for this outflow.