I found last night that the F-350 has a bad glow plug relay, as well as three bad glow plugs.
Left/passenger side, from rear of truck to front: bad (also had a bent pin in the connector, which was contributing to the problem); good; good; bad (so the two outers bad, two inners good)
Right / driver side, from rear of truck to front: good; good; bad; good.
I found eight supposedly genuine new Motorcraft ZD-11 glow plugs on eBay for $25 (if they are genuine, that’s a third of the cost) and a new Motorcraft DY861 glow plug relay on Amazon for $50. They’re on the way.