A lot of the real early work in computing was done by George Boole, before Shannon picked up where he left off. That kind of stuff (boolean algebra, information theory) will always be relevant... (to survival in ye enden times? maybe not) and it doesn't involve electricity.
Just as I imagine that the things that you learn as a speech therapist go way deeper than just helping kids with their lisps.
It's easy to get caught up in the current fads and stay on the surface (e.g. programming languages, or frameworks) of the computing field, and maybe what you're experiencing is a kind of existential dread about the meaning of the work you're doing. But I think if you stick with it and go deep, you'll find that "computing" is a fundamental aspect of the universe.
And if the lights do go out, then maybe everything you learned while "studying computers" will be of help in figuring out how to get them back on again... Or, better yet, help you to realize that you didn't actually need the lights in the first place.
Only things I see left in the "relevant" column by then would be how painlessly you get to go and your relationship with Jesus Christ.