Although I can't explain why the shift occurred, I found an interesting item in an 1839 Vermont newspaper that suggests that at that time and in that place some cultivated (or would-be cultivated) English speakers may have viewed the "so" pronunciation of sew as a vulgarism.
From "Vulgarisms" in the [Brattleboro] Vermont Phoenix (January 18, 1839):
Thr following list of unpardonable vulgarisms, were published in a New Hampshire paper, half a century ago. It was prepared for the use of a school, and it is said, proved highly beneficial.
These inaccuracies of pronunciation are not confined to New Hampshire, but exist now, as well as formerly, in different parts of New England. Many might be added to the list—and it should be the one of the most important duties of an instructor of youth, to do away with these barbarous provincialisms:
...
Shot or Shet for Shut.
Shear for Share.
So for Sew.
Sot for Sat or Set.
Sor for Saw.
Webster's Eleventh Collegiate Dictionary (2003) gives only one pronunciation for sew—the unpardonably vulgar 'sō—as indeed does every other Webster's Collegiate edition, going back to the first edition (1898). This would seem to suggest that the correct pronunciation of sew, according to the Vermont Phoenix, effectively died out rather quickly in the United States, between 1839 and 1898.
Further research on this point, however, makes the newspaper item even more dubious. Noah Webster, An American Dictionary of the English Language (1828) begins its entry for sew as follows:
SEW, v. t. pronounced so, and better written soe. {Sax[on] siwian, suwian ; Goth[ic] siuyan ; Sw[edish] sy ; Dan[ish] syer ; L[atin] suo.} This is probably a contracted word, and if its elements are Sb or Sf, it coincides with Eth[iopic] ~~~ shafai, to sew ; and the Ar[abic] has ~~~ an awl. ... The Hindoo has siwawa, and the Gipsey siwena. But the elements are not obvious.}
Webster was a native of Connecticut and would certainly have been aware of a different—and preferred—pronunciation of sew among cultivated New Englanders if it existed. Since he indicates no such awareness, I think the likelier explanation is that the objection to "So for Sew" in the 1839 Vermont newspaper isn't objection to pronunciation at all, but to phonetic spelling. If this is the case, though, it is striking that the newspaper views so as a barbarous provincialism at the same time that Webster endorses soe as a "better" way to spell sew.
[soʊ]. But sewer does not sound like sower.[sjuː].