Chosen Solution

2015 Macbook Air 11” (not retina) immediately shuts down when unplugged. What are my next steps?

Your battery still has quite a bit of life so I don’t see that at your issue. Lets try resetting the SMC to see if that helps How to reset the System Management Controller (SMC) on your Mac

Battery seems to be fine and probably not the source of the problem. If you had some current surges or an aftermarket charger that might explain the issue, but things may break on their own for no special reasons too. There’s not much you can do on a DIY basis, except trying a replacement battery, but I wouldn’t recommend that with those data. A logic board repair or replacement seem to be the two options in this case.