How do I improve my technique to reduce string sound?

I still have a level of string sound on roundwounds that is not acceptable for when I'm recording. I'm looking for advice!

Here's my issue: I watch what other players and my online instructors do, but when I mimic them I have WAY more string sound than they do. If I use my index finger to slide, I hear string sound. Walking up chromatically? String sound. I hear string sound during almost all movement of my fretting hand.

Here's what I've tried: I am doing my best to press more lightly on my fretting hand, which has helped some. But when I slide between positions, my unused fretting fingers (which are helping mute) always end up causing string sound when I move. So, I'm at a loss. I can either not mute, and get sympathetic resonance, or mute and get string sound.

Lately I've been using headphones when I practice to make sure I'm focusing only on what get through the amp, not acoustically. But I'm still struggling to make real progress.

BTW I'm not looking for ways to *hide* string sound (flats, eq). I want to improve my technique.

Thanks in advance!