The battery light comes on when the alternator is not charging. Changing the battery did nothing. Either put a gauge on the battery to see if it is charging, or while running, disconnet the positive cable from the battery. If the car dies that means the alternator is not charging. Most parts stores will check it for free. Oriely's , Autozone, etc.
check the fuse in the charger but your battery could need replaceing
An auto battery could be leaking for a couple of reasons. The alternator could be overcharging the battery causing the leakage. Or the battery itself could have a short or other internal problem causing it to leak when charging.
Battery could be dead , or a poor cable connection to the battery
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Dead cell in the battery or something is on drawing power from the battery.
Leaking from where on the battery, the caps? If so the charging system may be over charging the battery and causing the acid to boil out. Could be a voltage regulator problem.
The key being in the "on" position itself. As for rapid battery drain, you could have a bad battery or there could be a loose connector. Source: Personal experience
A bulging battery is an indication the is a short INSIDE the battery and it is getting overheated, causing gas to expand inside. It could explode on you.
The expedition door lock relay could be stuck causing the grinding noise. Check to insure that the lock relay is working properly.
Dead cell in the battery itself which requires you replace the battery. Can also be any light on the vehicle that is on pulling power from the battery. It can also be a relay that is stuck closed.
battery fuse is burned out, its in the fuse box inside engine compartment.
Some light may be on pulling power from the battery. Can also be a stuck relay. The new battery is possibly defective. Alternator can be weak.