Boeing, I had a similar problem and did the following. My 40 car has been converted to 12V except for the gauge cluster which still operates on 6V using a Ron Francis CONSTANT VOLTAGE REGULATOR that provides a steady 6V. Been using it for 15 years. However, when converting to 12V, I isolated the BATT (voltage) gauge from the original circuit and used an in-line resistor that reduced the incoming 12V to 6V. I did this:
1. Disconnect battery cables from battery
2. Buy a 6V dry cell battery from Lowe's ($4).
3. Disconnect both wires from fuel gauge and, if possible, remove it from your car. Makes it easier to work on.
4.Remove your fuel tank sending unit from your tank and gently clamp it in your vise so you can pry off the top with a brake adjusting tool or big flat blade screwdriver.
5. With the sending unit clamped in vise so that you can move the float arm, hook up the gauge and sending unit as shown in pictorial sketch. After removing the cover, you can attach the input wire directly to the brass spring arm that rubs against the inside of the top of the cover. No need to replace cover after each adjustment.
6.Then move the float through it's range of motion to see how it reads.
7.Using a small flatblade screwdriver, I turned the little star wheel CW to make the needle read closer to FULL and CCW to make it read less.
You can actually touch the bi-metal winding and feel that it gets hot based on the operating principles of the KS unit. I also checked to make sure that the carbon resistor inside the unit was fully seated with good contact. Also checked the soldered connections of the hairline wire. Also, clean all contact points with 600 emery paper.
After an hour or so of playing with the unit, I got the gauge to read better, replaced cover and tried it again to see if it was still ok. It read a little low so I again removed the cover and turned the star wheel CW an itsey, bitsey amount. Then recovered unit and it worked good.
It was interesting to do this and to actually see how the gauge and the sending unit work together on the principle of the bi-metallic strip - not OHMS.