I think strings will always resonate when others are played, all that vibration has to go somewhere.
In terms of sensitivity maybe your pups are a little high, I had to experiment quite a lot before I got a balance I liked, I love the mids my NB's produce when they are as close to the strings as I can get them, but the lows get too boomy, so I had to lower them a bit. You could try mounting them at a slight angle, a bit lower on the low E side. I haven't tried to mess with the pole heights yet, but you can do quite a lot just with the overall height.