Sounds like you don't have enough foam underneath the pickups which pushes the pickups up against the screw heads. As Line6man said, get more foam under there.
The distance of the pickups to strings will affect tone and loudness. If you double the distance between string and pickup, the magnetic field decreases 4 times (squared relationship), which decreases output to your amp, so you do want to pay attention to distance.
For a starting point, visit your bass manufacturer's website and see what they suggest as a spacing for your pickups - once you have more foam in there. From the suggested spacing, you can adjust slightly to balance the sound of the lower and the higher strings to your taste.