Using the EXES instrument on SOFIA, we have obtained velocity-resolved spectra of several pure rotational lines of H₂ toward shocked molecular gas within three Galactic sources: the supernova remnant (SNR) IC 443 (clump C), a protostellar outflow in the intermediate-mass star-forming region NGC 2071, and the SNR 3C 391. These observations had the goal of searching for expected velocity shifts between ortho- and para-H₂ transitions emitted by C-type shocks. In contrast in our previous similar study of HH7, the result of our search was negative: no velocity shifts were reliably detected. Several possible explanations for the absence of such shifts are discussed: these include a preshock ortho-to-para ratio that is already close to the high-temperature equilibrium value of 3 (in the case of IC 443C), the more complex shock structures evident in all these sources, and the larger projected aperture sizes relative to those in the observations of HH7.