First, let's TROUBLESHOOT.

Obtain a voltmeter, like one of those you plut into the cigarette lighter. Start the car, what voltages do you see? Voltages over 13 show the alternator is charging, less than that, and you need an alternator.
If the electrical system is causing the rough running (I doubt it is but IF), then the voltmeter will show it and the mystery is over.

There is NO LOAD in the car heavy enough to cause this to happen, so disconnecting the siren or whatever is not going to make the car run right - altho it might stop the battery from draining when the car is shut off.