You need a new battery too! MacBook Air 13" (Late 2010-2017) Battery and here's the guide MacBook Air 13" Mid 2013 Battery Replacement
As you can see the cycle count is over the expected cycle count of 1k Determine battery cycle count for Mac notebooks besides a battery life is about 4~5 years which clearly long since passed as this is the original battery!
FYI: I use my 2011 15" MacBook Pro system (has a 2012 logic board now) every day doing quite a lot, I'm on my 8th battery, just installed. I get about three years of life per battery.
Your errors are related to a old battery. Let’s build a mind model so you can see what’s happening. Your battery is like a toilet tank which holds water. A fill system and the flush! Once the tank is at the volume of water the fill system shuts off only to come alive after you’ve done your business. But over time the rubber seal which holds the water from exiting rots out so now the water leaks out! At some point the fill system engages topping off the tank. But over time the seal keeps breaking down before you know it the fill is constantly running! Your water bill shows the effect if you’ve spent the time monitoring it.
So how does this apply here?? Power is no different! The battery is the tank and the charger is the fill and your logic board the flush!
But unlike water, power creates heat! Your charger and battery heat up as the power runs through. The logic board senses the heat or the low battery voltage and responds! Pumping up the fan.
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