Electric guitars are sensitive to any string vibration. With high gain, this becomes a big issue. You have to clean up your technique. A new guitar will have the same problem, because you are causing it. You have to provide for muting the strings you are not playing , as well as fretting the one(s) you want to hear. Its a whole new level above what you can get away with on an acoustic.

As they say... practice, practice, practice...