Have you checked the system's charging voltage while the engine is running?
The problem just might lie there instead. You should find about 14.4V. If
you find closer to 12.65V-13V, the battery is not being charged properly.
Also check the battery's state-of-charge. Do it this way (engine OFF):
1) turn headlights on for 5-minutes
2) turn lights off again and wait 5-minutes more
3) connect multimeter across battery terminals.
You should find 12.65V.
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