Nasty overtones on my strat - please Help

Try changing the string, if you haven't already. That description of 'two notes' sounds extreme enough to be a bad string - loose winding or damaged core.

Check the truss rod isn't loose - you can tighten it slightly if necessary without affecting the relief, you just need enough to stop it rattling, if that's the cause.

Look for a burr or any other flaw on the bridge saddle.

 

your pups being Lace Sensors, don't apply any magnetic pull on the strings, so there's no need to lower them.

I would agree with John that something must be loose on your guitar and vibrates, if it isn't a bad string.

I had a similar issue twice on an Am. Std.
The first time it was a broken string core, which caused ghost tones till the string broke completely.

The second time, it was the neck tilt adjustment bolt on the heel of the neck that came loose and kept on vibrating, causing those ghost notes and buzzing.

Find what's loose and replace the string.

good luck

 

Also, check your trem claw springs. Mine used to vibrate and rattle until I stuffed cotton between them and the guitar body. The vibrations used to work there way up into the bridge and from there to the strings and pups.