The fuel sending unit is having troubles for two reasons.
First, the gauge wire ground is faulty and is a known problem with all the CJs.
CURE: Cut a hole in the floor of the bed over the spin out cap on the top of the gas tank, which is easier than dropping the tank, and pull out the sender, wire, and that phoney, engine stopping filter than gets the algea growing on it and stops our Jeeps in their tracks. Replace it all with a copper sender and solid wire dip to the float bottom. Screw in the cap and then make the cut out piece of the bed a cover with tapping scrrews so you can always get to the tank without dropping it.
Second, replace the gauge for $35 if it is old, and make sure it is wired for ground across to the temp gauge too--this is OEM. Mine works perfectly and always has since repairing it 14 years ago when I rebuilt VEX as a total.
I won't tolerate gauges not working and they should work perfectly and do, even my 4WD inidcator light and other APUs work perfectly as they should.
If you replace all wires as you go wth super heavy ones and grounds to match, your electrical problems disappear.
The rule is no electrical system is as good as the weakest ground, on 12 volt systems. So, beef it all up with the heaviest wire practically you can use.
I use 2/0 (not 2 gauge), welding cable with copper ends, to 2 - 1000 amp batteries in parallel (not isolated), powered by a 200 amp alternator. All my lighter wiring is 12 gauge, no smaller.
Strangley, lights, indicator lights, marker lights, etc, last about 10 times longer since there is no resistance to speak of in the system, under this arrangement.
I hate electrical problems!!!