No, you do not want to add more relief. You have about 10 times as much as you want. Tighten that truss rod more.
Folks always think they need relief to give the strings room to flap around, but the geometry doesn't work out like that. The maximum point of bow in the neck is near the midpoint of the neck, but the widest point of vibration of the string is not. Too much relief actually puts the highest frets closer to the vibrating string, causing buzz.
A setup has to be done in this order, because everything affects the items down the chain:
- Set relief
- Check nut slots
- Set action
- Shim if necessary to set action adjustment range
You can skip the nut for now. After you set the relief properly, set the action to the appropriate range (usually 4-5/64") and play it. If you have a straight neck and significant buzzing at 5/64" or higher, then there are high frets. The only way to fix high frets is a level & crown.
If you still find that the intonation screws are sticking up, then you'll want to put a shim in the front end of the neck pocket (towards the headstock). Start with one business card thickness (I usually trim them so they're about 1/4" wide and just long enough to fit between the neck screws).