How do I mute the other strings when playing power chords so it doesn't sound so mushy?
First of all, you need to work on your picking making it targeted only towards the strings power chords consist of, and avoid accidental picking of nearby strings.
You can mute the nearby strings by touching them slightly with any part of your available fingers on the fretting hand.
In this example the index and the pinky finger press the A and the D string for power chord performance, while the middle finger slightly touches the next unused low E string and the side of the pinky finger the next unused G string (the gray circles).
You can also use your thumb to muffle the E low string.
When you transit the fretting hand from one power chord to another located on the same or different strings, in the moment the hand takes off from the first chord, it "automatically" stops the strings from ringing. And if you play the next chord there won't be noise.
But if you need to pause before playing the next chord, like here:
the noise would be occur, so in the breaks between chords put the edge of your picking hand across the strings, like you'd do with
palm mute technique.