Why wont my 1990 chevy corsica start on its own?

You're not making enough power to run the engine and the lights at the same time - this sounds like the battery or the alternator. I know you said that they were both tested, but if the battery has an internal break (usually caused by overtightening the terminals, or sharp impacts), it can test fine and still not work when in the car - my wife's car did this to us last winter. The only other possibility I can think of is some sort of weird electrical glitch - and finding those can be very time consuming - and expensive.

Good luck!

The battery and alternator may be good but are the cables? Replace the battery cables. That should do it.

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Might have a ground issue something to do with the headlights.

try getting the computer tested