1998 Dodge Neon radiator and starter

Electrical problem
1998 Dodge Neon 4 cyl Automatic 150054 miles

Over the past month and a half, my car has overheated every week. A week and a half ago I took it to the mechanic and he changed 2 hoses, my thermostat and the gasket for that. A few days ago, I parked in my driveway and steam started rolling out of the front of my car. My radiator died. Yesterday, we bought a brand new radiator and replaced it. We went to turn the car on and it wouldn't turn over. The electricity came on, however, my turn signal was mysteriously blinkiing away and would not turn off. Even when the key was in the ignition enough to make the beeping sound when the door is open, the turn signal was giong. I need help. I wasn't having any problems at all with my starter or my turn signal before my radiator went bad.