We present Green Bank Telescope (GBT) observations of the 3(12)-3(13) (29 GHz) and 4(13)-4(14) (48 GHz) transitions of the H2CO molecule toward a sample of 23 well-studied star-forming regions. Analysis of the relative intensities of these transitions can be used to reliably measure the densities of molecular cores. Adopting kinetic temperatures from the literature, we have employed a Large Velocity Gradient (LVG) model to derive the average hydrogen number density [n(H2)] within a 16 arcsecond beam toward each source. Densities in the range of 10^{5.5}--10^{6.5} cm^{-3} and ortho-formaldehyde column densities per unit line width between 10^{13.5} and 10^{14.5} cm^{-2} (km s^{-1})^{-1} are found for most objects, in general agreement with existing measurements. A detailed analysis of the advantages and limitations to this densitometry technique is also presented. We find that H2CO 3(12)-3(13)/4(13)-4(14) densitometry proves to be best suited to objects with T_K >~ 100 K, above which the H2CO LVG models become relatively independent of kinetic temperature. This study represents the first detection of these H2CO K-doublet transitions in all but one object in our sample. The ease with which these transitions were detected, coupled with their unique sensitivity to spatial density, make them excellent monitors of density in molecular clouds for future experiments. We also report the detection of the 9_2--8_1 A^- (29 GHz) transition of CH3OH toward 6 sources.
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