I've been working on a lot of alternate and sweep picking, and as a result I believe I have fairly strong muting technique. One thing that's becoming a bit of a hangup is when I lift my finger off a fret (particularly the 7th or 12th) before immediately changing to another string, the note on the next string sounds terrible because the harmonic is still ringing out. I can prevent this with my picking hand thumb when the next string is higher, but when descending to a lower string, I can't think of a good way to prevent the harmonic from ringing. Any suggestions?
Sounds like you are playing a bit sloppy. Slow it down, make it smooth so every note rigs out clearly and you don't get an overtones.
I actually practice everything very slowly and strive for accuracy and consistency. The problem is that I can't seem to find the proper way to release a string when I'm playing over a fret prone to harmonics without actually causing one to ring for at least a moment, even outside the context of a phrase.
what I do is either slightly lift my finger to make the note 'dead', or I keep my finger closer to the last fret (finger on 12th fret is closer to 11th but without creating fret buzz), however, I only have a small practice amp so my guitar doesnt create as many harmonics as yours probably does.