Open it, remove the battery and plug the charger in (it will boot). Try the keys and if the issue is still there, the issue isn't the battery.
Check the keyboard connector on the trackpad board for liquid damage and also both sides of the trackpad cable. If there is no liquid damage on either the issue will be keyboard related.
Most likely it would be a keyboard issue, considering the shift key is not working at all. I think it is probably coincidental the battery is saying there is an issue (although you should replace this too, might expand in future).