The answer to your problem? Technique. Analyse your problem and experiment with different ways to resolve or work around it.
I think it probably has to do with the way you fret. Your fingers must be pendicular to the fingerboard, as least as close to pendicular as possible. When you lift your finger to another position, make sure you lift directly upwards from the fingerboard, and not in a curved fashion. The curved method (going slightly up or down along the fret as you lift off) is used for pull-offs to "ring" the string you're lifting from.
You can mute with your fretting hand or your palm depending on where that problem string is. It's kinda hard to explain, leh. Let's say you fret a G chord, then switch over to D chord; your low e string rings during the switch. You can use the palm on your picking hand to mute (dampen) the low E string. Say you fret a D chord and switch to C; your high E rings. You can then use the flesh of your first finger to mute the high E. Something like that.
Practice more! I bet you'll discover your answer on your own! Cheers!