It may also be a sort of mondegreen (though I can't think of what).

Consider that Tom is putting on the pathetic act, so the overt reading of doldrums (indicating drooping expression, drooping shoulders/stoop) suffices. Applying a measurement points out his exaggeration, and serves to clarify the secondary meaning, critical to explaining his welcome. As you suggest, it would not have done to say, "britches torn behind and before, exposing his prodigious dangler." But, for me at least, to find a merely prosaic meaning of "doldrums" would take away from this song.

Direct sexual slang is much more common in the living tradition than the records of it indicate, particularly with songs of this nature; consider the countless filthy lyrics people make up to any popular song. Many songs live double lives, coexisting in both expurgated and nixpurgated versions (the latter circulating only orally). Songbooks and recordings present only a pale reflection of the living tradition. The very oddity of a word like "doldrums" sometimes suggests a clean substitution made to stick out as a substitution--no need to look further.