I have a 1996 Taurus with 100,000 miles on it. The engine is the standard 3 liter v6. I have taken it to the local dealership for an electrical problem that is vexing. Mostly it starts and runs well. But every once in a while it starts extremely slowly, as if the battery were completely (or very nearly so) drained. When I turn the key it almost doesn’t start, yet it finally does pull through. The next time I start it, it is just fine. The dealership tested the starter motor, the charging system (it has a new fan belt installed last December, and a new battery.) and they both checked out fine. I suspected the new battery and took it back to Sears and they gave me another battery. Same problem! Obviously I don’t want to throw gobs of money at the problem, since the car isn’t exactly worth gobs of money. But it is a good car and I’ve never had any problems. I should note that about ten years ago I had a bizarre problem. The headlights had an aftermarket switch installed that would run the headlights whenever the engine was running. Then one time the lights refused to go out. It seems the car had been in an accident and the left front fender replaced. The mechanic indicated there was a short but he couldn’t trace it. The solution was to remove the fuse for the aftermarket switch (the lights work fine without it). And I haven’t had any difficulty with the lights since then, and that was 10 years ago. I don’t know if this could be part of the problem, but wondered if it is possible.
Get a voltmeter so you can check the battery voltage the next time the trouble happens. Check the battery voltage before you get the engine started. See if it has a voltage more than 12.5 volts. There may be something that is draining the battery while the car is parked. If there isn’t a drain then there must be a problem with one of the cables to the starter. There may be a intermittent problem with the starter also.
The accident could have something to do with it. If it was severe and it bent the body at all, it could have affected the way the starter motor is grounded.
That’s all I got.
Good Luck!