This is a common thing with newer players learning the muting process. The thing to practice is remembering that you have two hands to mute with, not just your palm. The fingers on your fret hand can mute open strings and so can the extra fingers not used to hold your pick. It just takes a light touch from either (or both) direction(s) that is comfortable to mute unwanted noise. Focus on these techniques and it will become natural over time, just like tying your shoelaces.