OK, just n case someone else has this problem, the issue was the power cord. I had another cord that fit, so I was using it. When I tried the one that came with the phone, everything worked great.
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I tried to upload to my computer yesterday and wasn't able, so I wondered if that was the problem. I'd never heard of it effecting alarm, but who knows, right!
Do you get other phantom touches when the phone is plugged in and charging? E.g, if the keyboard is open, do you ever have issues with false keypresses?