Battery light randomly comes on

You have a ground problem. Since it starts battery to engine ground is good. You need to check or add ground wires from the battery to the body, from the engine to body, and I put an extra ground wire from the transmission to the engine.

Your battery is not charged and gets drained constantly, dome lights doesn't take much voltage to be lit. Get a voltmeter and check on the battery. Engine off it should be all charged up and 12V ish. Try to start your engine, should be 13-14v charging, if not, then it is your alternator not charging or SOMEHOW your battery doesn't receive enough charging from the alternator.
When you turn on the headlight while engine is running and died off because the alternator some how couldn't supply enough current to power both headlights and motor ignition system
Get your alternator tested, before that all connections between alternator and battery is NOT LOOSE, or broken or damaged. Then go to auto electrician. Then sort out your fuses later.

what speed are you driving at, in 4wd ? before it does this ?

it sounds like that might be a short in the wiring somewhere, I'm not an expert on cars and trucks, but being a computer geek, usually when something pops on randomly and or stays on unexpectedly, it tends to be a short in wiring somewhere, however I would recommend getting a second or even a third opinion on this.

If the battery light comes on while your are driving than the alternator is not charging. The battery light comes on to tell you there is a problem with the charging system not the battery. It is normal for the light to come on when you turn the key to the run position and then it should go out shortly after the engine starts. When the light is on check the voltage at the battery if you have less than 13 volts the alternator is not charging.

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Sounds like a shorted or sulfated battery to me.

Do you have any codes in the engine control module ? When the vehicle shots off, do you think it's an electrical issue ? Of a fuel problem ?
Why did you replace the battery and starter ?

the ignition switch itself,or the ignition relay is bad.the ignition relay is in the fuse box under the hood.swap it with another relay with the same number.another thing you could do is run 12v from the battery to the pin that is activated by the relay when it is turned on.This will power up the cluster and if the car stays running fine,the ignition switch or the wiring from it to the relay is bad.The crank sensor failure will set off a check engine light as the pcm will detect a variation in engine rpm and set a p300 random misfire code,or the code for the crank sensor.

Its the transmission shifting sensor its located on the right side of engine near the bottom its a common problem its only like 45 bucks at dealer

engine light keeps comeing on n sometimes the battery light comes on my 94 lexus es 300 wat does tat me