Fluctuation in resistance value
Unfortunately, what you show here ... 



is not a 4-wire resistance measurement of the screw. it's a 4-wire measurement of the resistance of the screw plus the two wires to the screw, plus the contact resistances of the wire to screws, and wire to proto-board sockets. Of these, the contact resistances will be the most troublesome, as they vary hugely with pressure.

What you need is four discrete contact points actually on the screw, like this



This measures the resistance of the length of screw between the voltage terminals.

This way, the voltage drop at the current terminals doesn't contribute to the voltage measurement, and there's no current flowing at the voltage terminals.

You can interchange the current source and the voltage measurement if you like, the resistance measurement is still of the length of screw between the closest pair of terminals.

If you want to measure the whole length of the screw, be careful that the voltage leads and the current leads do not touch each other, as then a voltage connection could pick up some of the current lead voltage drop. Perhaps touch them on opposite sides of the screw.

Once you have four contact points on the screw, it doesn't matter if they are rusty or poor, as long as your current source can keep the current steady enough to measure the voltage.

There are limitations to how low a resistance you can measure, milliohms should be possible with this setup, as long as you have enough current available. 

Unfortunately, once you are down into the millvolts of voltage drop along the screw, another error phenomenon starts to rear its ugly head, offset voltages generated at the voltage pickoff contacts. These result if the object and the probe are dissimilar metals, and vary with temperature and moisture levels. These can be mitigated to a large extent with AC excitation and measurement, or reverse the current source wires and average the two measurements, or subtract a zero current measurement.