2001 Cadillac DES

I have a 2001 Cadillac DES. I was driving on the highway and the headlamps began to flicker on and off, same thing with dash lamps and warning lights, electronic speedometer went crazy. Battery said low volts. I got home and turned the car off and went to re-start it and it would just click. I took the battery and had it tested. Tested good. I noticed when I turned the key off the fuel pump would continue running. Unhooked battery cables then re-hooked them and the pump was off, I turned the key over and off again and pump did same thing. Possible Ignition problem? Any help is greatly appreciated.