The sensitivity to the vaporization Damkohler number of the behavior of a coflow laminar spray diffusion flame in an oscillating flow field is investigated. Droplet grouping induced by the host gas flow oscillations is accounted for, and its effect is described through a specially constructed model for the vaporization Damkohler number that responds to the proximity of the droplets as they cluster. A formal analytical solution is developed, and the dynamics of the spray flame front shapes and thermal fields are deduced. Computed results demonstrate how strongly the vaporization Damkohler number impacts on the type of primary homogeneous flame formed and on the possible existence of multiple flame sheets as the flow field oscillates. In addition, isolated regions of high fuel vapor concentrations are produced by fuel droplet enrichment. The presence of resulting parallel fluctuating thermal fields indicates a potential impact on undesirable pollutants production.