Molecular line observations are reported of two regions containing small cometary globules at the edge of the Rosette Nebula. Observations of the CO, 13CO and 18CO J = 2 - 1,and CO J = 4 -3 molecular lines towards Globule 1, the most prominent of the group, show it has a well-developed head-tail structure, with a head diameter ~0.4pc, and a tail extending ~ 1.3 pc behind it. The major axis of the system points about 45 degrees away from the direction to the centre of the Rosette Nebula (which contains the presumed illuminating stars), and 20 degrees out of the plane of the sky, along a projected line towards the luminous (924 L⊙) infrared source IRAS 06314+0427. The CO lines have a complex velocity structure; with a pronounced broadening at the front of the head (as viewed from IRAS 06314+0427); a velocity gradient ~1.4 km s-1 along the tail, and material at the front of the head is blue-shifted by ~0.5 km s-1 compared to surrounding gas. The CO J = 2 - 1 line intensity peaks towards the front of the head, and along the edges of the tail. The 13CO J =2 - 1 antenna temperatures in the head are very similar to those of CO, suggesting very high opacities or column densities, or that there is significant CO self-absorption. There is a narrow rim of CO J = 4 - 3 emission around the front of the head over a limited velocity range, which correlates with the position of a faint optical rim, and a narrow ridge of 2 μm H2 emission. These data give strong support to the Radiation Driven Implosion (RDI) model of Lefloch and Lazareff (1994 - hereafter LL94), which was developed to explain the physical structure of cometary globules. Using an RDI simulation, a remarkably good fit to the data has been obtained, allowing the CO, 13CO and C18O spatial structures and velocity field to be modelled. This simulation suggests that Globule 1 is ~400,000 years old, and has a mass ~ 50 M⊙. Additional observations towards the region close to IRAS 06314+0427 show that it is associated with an intense molecular concentration lying at the northern end of a ~ 5 pc long molecular ridge, with a mass ~ 330 M⊙, and lies close to the centre of a shell-like condensation
