Hi Andy, after reading the above I suspect you may just need to down tools for a bit and read up on setting intonation. Leave your neck as it is for now. (and order a new nut)
The saddles are the 6 little adjustable metal things that sit on the bridge.
Basically this:
1. Tune your string open
2. Fret the string at the 12th
3. If sharp (high) then the saddle needs to move towards the back - so tighten the screw
4. If flat (low) then the saddle needs to move forwards - so loosen the screw
5. Repeat these steps until your string is in tune both open and at the 12th fret.