Hey Ken,
Well if you're hearing it on particular strings when played acoustically, then your problem certainly is
not the ES! The answer lies in the nut or frets:
- If the offending string is
played open and you hear the nasty overtone, it's likely a nut cut too low. It's buzzing, albeit just enough to cause your tonal malady. If you play more lightly on the same string, does the ugly overtone go away? If so, it's because the string is vibrating in a smaller arc so not hitting the frets. The offending strings' nutslots need to be filled and recut less deeply.
- On the one bad string, does it occur all the way up the neck on every fret? If yes, it could be as simple as adding a bit more neck relief to allow the strings more room to vibrate cleanly. Neck relief, and possibly in concert with a minor change in the neck angle (authorized-Taylor tech ...easy peezy, possibly even covered under warranty), may yield a cleaner fundamental and eliminate said overtones. I know you had already mentioned relief, but at the risk of sounding obvious, the "correct" neck relief is that which works for
your style of play. And your break angle is simply
not going to create any overtone/pinging (unless it is egregiously shallow, which on an acoustic-guitar saddle is frankly pretty dang hard to accomplish!).
- As you said you get the bad overtones on fretted notes, are they on
every fret on that string? If not, then it's possible (likely) some of your frets are high ...in other words the frets need a proper leveling. This is common, and obviously is part of the setup process at the factory before letting it go. Perhaps it slipped through QC and you've got a less-than-ideal fret level. Perhaps not "bad enough" to make the guitar sound like crap, but not good enough that some fretted notes fail to ring as cleanly as other fretted notes.
Hope that gives you a few things to try and/or look at.

Edward