Sadly, your system needs some deeper TLC here ;-{
While you could just replace the logic board on your own, you’ll also want to replace the battery as well.
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* [guide|90276]
You’ll need to think through the costs. Do keep in mind this is a window to upgrade the CPU and/or RAM as well! It all depends on what your current system offers.
If you have access to good Mac repair person nearby then they should be able to fix things too!