We present Green Bank Telescope observations of the 312--313 (29 GHz) and 413--414 (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'' beam toward each source. Densities in the range of 105.5-106.5 cm--3 and ortho-formaldehyde column densities per unit line width between 1013.5 and 1014.5 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 312--313/413--414 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, makes them excellent monitors of density in molecular clouds for future experiments. We also report the detection of the 92-81 A -- (29 GHz) transition of CH3OH toward six sources.