Determining the sensitivity of model outputs to input parameters is an important precursor to developing informative parameter studies, building surrogate models, and performing rigorous uncertainty quantification. Determining parameter sensitivities over a range of parameter values, termed global sensitivity analysis, requires many model evaluations sampled over the parameter space, which is intractable for many large-scale computational fluid dynamics (CFD) applications. For moderate parameter dimensions, we propose the use of Morris screening, a one-at-a-time (OAT) quasiglobal method for estimating parameter sensitivities over a range of parameter values for CFD simulations. The Morris method is implemented within a CFD framework that utilizes adaptive grid refinement, thereby enabling its application to state-of-the-art production-level problems. The method is shown to be viable for a model problem of supersonic flow over an axisymmetric capsule geometry with both physical and nonphysical (i.e., simulation-informing) input parameters. Additionally, the feasibility and effectiveness of warm-starting simulations for sequential perturbed samples are investigated for this supersonic model problem. It is shown that Morris screening identifies relative parameter sensitivities consistent with those of more costly experimental and computational studies that were previously performed on this geometry