When you first put your hand down, it should be perpendicular to the strings, not laying flat against them. Slide the hand back as close to the bridge as you can– this makes it easier to learn.
It is often easiest to start by just picking one string. As you get used to keeping it palm muted, play a two-string chord, then a three, and onward.
With dampening, you should be able to hear the pitch when you pick the string. Try picking the strings one at time to make sure you can hear all the proper pitches rather than a dead plucking sound. [5] X Research source
However, as noted, you should keep your hand as close to the bridge as possible if you still want to hear pitch.
You’ll be lifting from your wrist, rather than from your fingers. You can curl your wrist up and down to open up or mute certain strings with some practice.
For a more distorted, buzzy sound, use the neck pickups.
Learn to start palm muting, then slowly raising your palm off. Become your guitars volume knob, switching fluidly.