Hemming and hawing

I spent some time yesterday hemming and hawing. At least I thought I did. I've always understood hem and haw as a reference to vacillation, to going back and forth in one's head — to buy or not to buy, to go or not to go. But whence this odd expression? I guessed at an explanation: to haw might mean to unstitch. Hemming and hawing might thus be endless doing and undoing, as if one were hemming a garment and taking out the stitches. I didn't hem and haw before deciding that here was a pretty plausible explanation. But as you may already know (or else are about to learn), hem and haw has nothing to do with vacillation or sewing. World Wide Words explains:

In Britain, we know it as hum and haw. Either way the phrase contains a pair of words that are imitative. A close relative of the first of these is ahem, indicating a gentle clearing of the throat designed to attract attention; hem more often represents the slight clearing of the throat of a hesitating or nonplussed speaker. Haw is very much the same kind of word. . . . In the British version of the phrase, hum is another word for a low inarticulate murmur. Either way, the two words together illustrate very well the hesitation and indecisiveness to which the phrases refer. There are other versions and both are closely related to um and er.

The OED confirms that the words are imitative and suggest hesitation in speech, not inner debate. I wonder whether my — er — misunderstanding of these words is a common one. Anyone?