Chevy Cavalier wont stay running....

sounds like your losing 12 volts in the run mode to the systems that need it. You really need a wiring diagram that shows where the 12 volts should be when in the run mode. You could be losing 12 volts to the ignition or engine computor in the run mode. Might check the ignition and pcm fuses. Some GM products also have 2 small wires that go up to the key cylinder lock that lets the computor recondise the ignition key, if one of these wires is broken,it will not stay running.