Today I replaced the ignition switch, sans the ignition lock mechanism and lock itself, which solved the problem I had been having with the wipers, horn and radio not working after starting. Unfortunately, this has led to another interesting development. What I'm guessing is the car alarm now goes off whenever I don't have the headlights on, after starting. I know the pervious owner had the alarm disconnected, so I'm not sure how I've activated it again. The horn or siren does not sound, its just the lights blinking. I thought initially it might be the hazards, but the hazards use the rear turn signal lights, while this problem manifests itself in the regular rear running lights.
I'm thinking I might have jostled the wiring while rooting around behind the dashboard, or that the newly working ignition switch might be activating a connection that was dormant before. I don't know if the car alarm is aftermarket or not, but it might be based on the fact its not mounted and just sits atop the lower dash trim.
Any and all suggestions appreciated.