Can you change a hearing aid battery on Shabbat? I read on this site that it’s okay to carry crutches outside of the Eruv if you can’t walk otherwise without them. So shouldn’t you also be able to change the battery if you can’t hear otherwise without it?

Carrying on Shabbat is permissible even without an Eruv if the item is in a way part of your body or an essential need (under certain circumstances) - like clothing, that's the reason crutches are permissible in the above case, it’s not that the laws of Shabbat do not apply to disabled people. Changing a battery is an electronic action that is prohibited on Shabbat for the connecting of the electric circuit in the device.