Since adding distortion when I release a string I can hear it, almost even sounding like a pull off, so now I've had to slow down and when I release I have to mute it with my fretting hand before moving on to the next pitch. Is this how it's done or is it better to do it with my picking hand which seems more difficult at this point. I'm talking about when I'm practicing scales.
This is a common problem, and being able to effectively damp notes after they are played is, to my ears, one of the differences between a decent amateur guitarist and someone with real control of the instrument. In fact, it is a good idea to damp notes when playing clean, too, although many players don't recognise the need until playing with distortion, which highlights extra unintentional sounds.
Yes, a good way to stop strings sounding after you lift a finger, is to do what I call "self-damping" notes, where you lift a left-hand finger a little slower, so that it damps the string it lifts off (this is what you describe). But, I find I also damp with the right-hand, too, particularly when playing scalic passages and/or with distortion. I do this by keeping my thumb parallel with the strings (pointing towards the headstock), so that it touches and damps strings as I pick each note.
Also, damping open strings, when playing with distortion, can help to avoid extra noise and control feedback. You can do this by touching strings either side of the one you're playing with the left-hand fingers, and by putting the heel of the hand (near the wrist) on the lowest strings, if playing on the higher strings.