Be sure to test the power supply before going too far down the rabbit hole.

You should be able to sign in to just the store app by clicking a tiny little box on the subsequent window that pops up during the sign-in process. That's how I sign in to the store for app downloads without having to use those same credentials to sign in to the PC.

That box is intentionally made small and easy to miss because they obviously want you to use a MS account across the whole machine, but it's there if you move slowly and look for it.

That said, you shouldn't have to sign in to download any app updates from the store. You can also download new apps without signing in-- the license will just be tied to the machine instead of your personal account if you do that.

This unhelpful error is just MS being MS. Best option is usually just to leave it be and check back later, as it oftentimes really is a problem on their end.