Big sigh. Newt's fuel gauge was obviously broken. I have a 100,000 mile warranty on her, so I put in a call to the service department here in town and made an appointment for the following Tuesday. The very next day, a Friday, I started Newt up and saw the fuel needle drop a tiny bit. The next day, it dropped down to three-quarters full.
I was a little embarrassed about the whole Newt is Already Broken thing, but since then, I've been reading my truck's manual. Like they say, read the directions first!