After so many kind suggestions, I just wanted to follow up on this in case it helps someone else. The unwanted sound has been gone for about a week now and hopefully will not come back. I noticed the bridge was tilting backwards a little towards the tailpiece, so shifted it a little forward to a more vertical position. This had the effect of increasing the string tension slightly, and the E string no longer made that sound. All the other strings sounded a bit fuller too, presumably because the new bridge angle allowed them all to vibrate more consistently on a more stable foundation. So the problem was not with the construction of the string, but apparently with the set-up.
