Interesting take on this as I have this one bass (only one) that when my finger hits the string behind it as mentioned above I get clicking sounds. I thought it was the bass and the action, and I posted in the setup forum but I have now isolated it to this cause. I'm considering changing strings because it is annoying and only happens with this one bass (I have 7). Has this ever happened to anyone else? The only thing I've found (so far) that alleviates it is to play extremely light which is hard to do consistently.
Try lowering the PUs. Sounds odd.
As for the OP... All I can say without seeing your style is that light touch ON BOTH HANDS is required for a nice, even, round tone.
Try this. Put your fretting hand index finger on 5th fret A string (D) and middle finger on 6th fret E string (A#).
Don't push them down yet, just keep the fingers on the strings. This is your starting position.
Put your plucking hand thumb ON the PU, like, not above it anchored. Just hovering on top of the PU, same side as the strings, perpendicular to the face of the bass. Idea with your thumb is to move back and forth between the area above the E string and resting ON the E string. Just for like 1cm or 1/4 inch whatever.
As for the pluckin' fingers, when you pluck A or E string, they should snap back to your thumb. Just roll 'em over the string and keep your LAST (first if you start counting from the fingernail) knuckle loose so it bends ever so slightly and snaps of the top of the string to your thumb.
Your objective here is control your fingers completely.
So on to the practice. Keep your fretting hand fingers ON THE STRINGS all the time. This means that when you lift off from the note you still keep the finger on it to mute it and you have to curve it a bit so the A string can vibrate freely behind it. It also means you have to keep your hand fairly relaxed so they won't fly everywhere.
So fret the E string on 6th fret, keep a beat with your leg/metronome slooooow tempo and watch your hands! Play 4 quarter notes while alternating your plucking finger and snapping to your thumb and for the next bar do the 5th on A string for 4 quarter note. For now don't get frustrated if it's not RIGHT on the beat, just concentrate on getting it fluent and relaxed.
This way you mute the A string with your fretting hand and E string with both the thumb and the fretting finger. In fact, you could play the same thing whilst pressing the 6th on E the whole time and it will be automatically muted with your plucking hand thumb.
If you wonder what your hands should look like when playing do this.
Put BOTH your hands in front of you palm facing the ceiling. Now turn your plucking hand elbow upwards towards the ceiling so your palm falls a bit and looks relaxed. See that space between the thumb and index finger? Insert strings there and keep the hand as it is.
Fretting hand? Still got it palm upwards? Pull it back to your body where the neck of the bass lies and insert the neck between the thumb and the fingers. As relaxed as you can.
If this helps, good! If not, find a teacher who can actually see what you do and can give you exercises to get your fingers goin'!