I'm not a great mando player and I struggle with this some, too, but I generally mute the E string with my edge of my index finger - almost like barring the second fret only without so much pressure on the E string.
When I switch chords, I only lift the fingers necessary and I usually don't lift them very far from the strings. For example when switching from a C to a G chop chord at the second position, I don't move my index finger at all, so the E string never has a chance to ring out. Similarly, If I'm moving up from a C chord at the second postion to a D chord at the fourth position, I'll release the pressure on the strings (but not lift my fingers off the strings) and slide to the new postion. In addition to stopping unwanted notes from ringing out, this economy of motion makes it easier to play fast, too.