2003 Oldsmobile Silhouette rear hatch

The 2003 Olsmobile silohoutte keeps the headlights on during the day even when knob in off position. The dayrunners are usually the only ones on. However they do go out when the car is turned off. The back passenger side tail light has gotten a 1 inch hole in it which once discovered because of taillight being out, had caused much corrosion to the top lightbulb socket. The whole socket piece and lightbulbs have been replaced. That tail light continues to only have lower half working unless brakes applied, then all bulbs work as should. Any connection to the two and any other suggestions to try?