Sounds like you just need a new battery... if the car starts when you're charging, the starter is most likely just using the power from the charger (or the charging battery). And if it continues to run once you've started it, then the alternator is fine.
It's possible that when you installed the new lamp socket that you created a short in the battery, or that the battery is just old and needs replacing... but replacing it is definitely where i would start.
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