Everyone has summed it up very well, not much that I can add.

When you pick the left hand up to move it, make sure you mute with the palm of the right hand. This would probably be the way to go, as it's very hard to block strings while your hand is in motion.

There will still be some noise, especially on an electric guitar. It may help to figure out when it's happening, and then work on ways to fix it. Slow down what you're doing, and see which of these options gives you the least string noise.

If your hands get really sweaty when you play, this can limit your ability to move up and down the neck smoothly. You may have to change strings more often, or use olive oil to make it easier to move your hand on the strings. I have used that trick to get another night out of old strings I didn't have time to change before a gig. You can get things like fast fret, finger ease, etc. I quit using these because of the toxins found in them. The fast fret isn't bad, and will last a long time, just make sure you take care of the container to store it in.

I guess I did have something to add after all, hope you figure it out. Let us know how it goes, and have a good weekend.

JD