yep. i vote dead/failing battery. (in the trunk btw. some people have trouble finding it. sometimes even dealers.
) the LS is
extremely voltage sensitive. charge the battery completely with a battery charger and try again.
do the tach and speedo make full sweeps after trying to start the car?
afaik, the PCM doesn't control lighting. that is dealt with with separate modules in the car, including, but not limited to, the REM (rear electronics module) and FEM (front electronics module).
a failure of multiple systems is unlikely, so there will be an underlying issue causing this.
other things to check, fuses in all 3 i believe it is, fuse boxes. check grounding points. corrosion on the battery terminals. any sort of wear or chafing of wires. etc.
note, if you do decide to replace the battery, it is strongly suggested you replace it with a sealed, tube vented motorcraft battery. as non sealed units can leak harmful gases into the cabin. and they might explode.