Ken,
Had a thought about your Kenmore and modern Singer needles. Modern Singer needles have a tiny punch deformation on either side of the top of the needle. "Sometimes" these little deformations will not fit in the needle grove of older machines and cause an out of alignment situation. After cleaning the needle grove in the needle bar, try a different brand of needles. Schmetz or Organ are good ones to use.
I have two of these machines, different model numbers but the same design made by White and they both use the standard 15x1 needles. I can't remember ever using a modern Singe needle in them, so what I've just said is theory to be explored.
Morningstar,
Sew-Classic lists walking feet attachments for the high shank machines.
Joe