I have not seen much electrical problems with the LS 400 apart from the instrument panel needles and heater control display. You need to ground the sender unit to gauge wire at the gas tank end and see if the gauge works and reads full with the key on. If not you need to check for power at the gauge/ instrument panel. I believe the fuse in the fuse box is marked INSTRUMENT or INST. There are two companies that independent auto repair shops "buy" their information from that offer a very reasonable service to do it yourselvers. You can get a one week subscription or year subscription very cheap. The information is year make and model specific repair procedures, service bulletins, component locations, wiring diagrams ect.... great quality information same as the professionals have,much better than generic workshop manuals from parts stores and much cheaper than the factory manual try Alldata which is very easy to navigate http://bit.ly/AllData_Repair_Manuals_Online or Mitchell on line http://bit.ly/Mitchell1_Online_Repair_Manuals and get an online repair manual subscription. Autozone have some popular workshop manuals for free on line once you register (for free) on their website http://bit.ly/autozone_repair_info.
capacitor in instrument cluster goes bad also. gives inaccurate and false readings. takes awhile for it to come on sometimes just like the dash illumination problem. A lot on the internet about this. replaced my capacitor for about 3.00 and an hour of easy work..