1992 Ford Taurus Engine shuts off intermittently while driv

I have a 1992 Ford Taurus SHO. Sometimes while driving or stopped at a stoplight, the engine will die with no warning. When it shuts down all the dash lights come on as if I put the key in and turned the ignition to the on position. After the engine dies I try to start it back up but it cranks strongly but will not fire. I usually have to wait from less than 1 minute to an hour for it to fire up.
Also eveytime I start the car up the battery light stays on until I rev the engine to 2k rpms at which point it goes off and does not come on again until the engine is restarted. When I rev past 2k rpms, the interior lights and headlights get brighter and I notice the fuel pump gets a little louder as if more voltage is going through the system.
Could a bad alternator be the cause of all this or do I have some faulty wiring? Ive been told it could be my crank sensor but the check engine light does not come on after engine is started. There are no codes. What could be causing this?