starter problem

Have a 1994 Gran Prix, 3.1 L auto, 124500 miles, The car had trouble starting and when it did start it wold not shut off even after the key was removed. Pulled the fuel pump fuse to kill the engine. then every 5 to 10 sec. the car would try to restart. Pulled fuses to stop this. Let sit for 2 days and replaced fuses, checked ign. swith to be sure it wasnt stuck anmd tried to start car. Starter solenoid hammered hard and no start. Replaced starter and works but hammers a few times before catching. Have tested battery and alt. both test good. thanks for any advise. Jim