My first thought was a grounding issue. My 2003 Ford Ranger uses a single flasher unit for turn signal and hazards however there are two circuits in that unit. In my case the turn signals would flash once and stop but hazards worked. Once I replaced the unit both worked. So I would not eliminate that component based on it working for turn signals and not hazards. That does not account for the brake lights. If they both were working pre plug change, I would go with my initial thought that it is ground related. That when you changed the plug a ground was disturbed.
Mike
