How often do you play loud volume/high gain? This is definitely a skill that is "picked up" over time, and playing in that kind of environment will help you out.
One definitely needs to "learn" to play loud. If that makes sense.
It can be tackled a few thousand different ways, as I'm sure we'll see in the answers here but what has worked for me is muting, since inevitably, we'll have problems with strings being ever-so-slightly plucked as you try to quickly lift your fingers off of them.
So learning to mute can be taken from a few different angles.
This assumes you're playing right-handed guitar. Easily think in reverse if this assumption is not true.
Assuming you've been playing for some time now (which you have) I can assume you're left hand is pretty good with muting. It may even be subconscious, but you'll need to make sure that an unintended glissando is not apparent (moving up or down the neck on a single string) in your playing. An easy way to work on this is playing at stage volume (whatever you consider that being) and really focusing on what's happening between the parts you're playing where a position switch is necessary. It may even be beneficial to record a few sessions and listening to the extraneous noise you encounter so you can focus on when and where it happens.
The other muting most of us are used to is the left hand finger's muting the strings as you lay them down. This is picked up pretty easily with enough volume and gain in practice, but it's always good to see how the interaction between horizontal (sliding) and vertical (laying down hand/muting) positioning works, and if there's any minimize-able gaps between these.
The right hand was really the Eureka moment for me as I started to play louder amplifiers, since we'll inevitably get finger/string noise. I practice a combination of left and right hand muting to really get a noise-free (or, as much as possible) performance and find a sweet mix of technique to properly play high gain.
With the right hand, you can easily rotate your wrist out towards the audience and get a nice fatty, fleshy (hate those words - better adjectives for that?) part of your right hand on the strings. Depending on where you pick most of your time, this will be more or less effective with either a full-stop mute, or a palm-muted sort of sound. The goal is to come up more on the string to effectively mute the noise you'll get from the string being amplified 10,000 times. I've found that with practice between this and the left hand traditional styles of muting, a lot of gain and volume can be played and still come out "cleanly".
A noise gate or compressor with a threshold can help slightly, assuming you have the cut-off point set well enough. The noise gate will allow you to "tune-out" some of this noise with proper muting technique applied, but it can have an overall "choppy" sound to it. The compressor will work in reverse of this, but bring up the overall volume of sound that's not the string noise making it a bit less apparent. But you really shouldn't focus entirely on this solution as it's gear-dependent, especially since you've admitted you're not playing with extreme high gain.
This is out of left field, but how often are you changing strings? Dirty strings can have more funk on them, making them a bit rougher and allowing perpendicular vibration to happen more easily (think of a violin bow across a violin's strings). A newer set of strings are "slick" and inherently allow you to slide across them more easily; you'll find the noise as you move across them lower than an older set that has maybe started to corrode or "die". If you find that your technique can not be improved, a look at the strings and their state of affairs may be in order. We're talking about unwound strings, but this even slightly applies to wound (although some may make the argument that gunk in the wound strings makes them smoother).
It's important to understand that some of this sound is intended, and subconsciously requested by many. It keeps you from sounding like a keyboard and that's a good thing. Acoustic guitarists can't always get away from this sound, and it's better to embrace it at times.
However, I understand the likeness of this sound is bipolar. Either people can't stand guitar because of it (and don't even realize that's the sound they hate) or can't get enough of it. It's good to find a nice mix of silence and this human aspect of playing the instrument. I mean, pick slides are the reason a lot of guitarists picked up the instrument in the first place!
My picking hand causes some of these noises as well but I think the problem there is playing too cramped when playing fast and having a tendency to rest my hand on the strings too much.
...You definitely should learn to relax your hand if you intend on playing for a long time. It allows you to mute strings a bit easier, possible, but more importantly it's something too many guitarist don't focus on until it's too late.