I have a 96 toyota camry that will start and run but the starter won't shut off.
When I turn the key off, the car engine will turn off but the starter keeps running. I
have to remove the negative cable on the battery to stop the starter from running.
Sounds like possibly an ignition switch issue.
i am going to have to say your starter solenoid is fried. it is the cheapest easiest thing to start with.
starter solenoid is arcing at contacts and fusing together and feeding starter. until cooled IE - removing bat.post
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