Car won't crank! What's wrong with it?!

I have a Ford Taurus. I was driving one day and everything started losing power, headlightsd dimming, etc. I pulled over and killed the engine. It wouldn't crank back up until somebody jumped it off. It barely stayed charged long enough for me to get home. Mechanic told me the alternator was out. I replaced it. Car wouldn't start. Had my battery tested and was told it had a dead cell so I bought a brand new one. Now my car will keep turning over but still not cranking! What could it be? Could a coil pack have been damaged from the battery being set on/off the top of it so many times?

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