Small update. I changed the filter tonight. After changing the filter, I bumped the starter many times to re-prime the system, and I definitely heard the lift pump spinning for maybe 20 seconds after each bump. A ticking noise came from the engine, which if I had to guess, I would guess was air being pushed through a check valve or something out of the fuel system. The ticking got faster and faster each time I bumped the starter, and finally the ticking stopped.
If you are curious, here's what the ticking from the engine sounded like:
BTW, the engine is the fast-paced ticking. The slower ticking, around once per second, is my electric fence driver. You can ignore that.
Anyway, when that all was done, I tried to start the truck. It cranked like three seconds and then the battery died. Given that it was on the trickle charger this morning and was in "float" mode, this is totally unacceptable. When the truck is running the volt-meter shows 14+ volts, so I am pretty confident the alternator is charging. The diagnosis is now looking more and more like a pair of old batteries.
Just to settle my mind about the "water in fuel" situation, I also dumped some Sea Foam in the tank. I have a 30 gallon tank, so I put 2 pints in (1 oz per gallon). After doing that, I realized that there's only about 5 gallons in the tank, so I quick ran to the Internet to see if too much Sea Foam was a problem. Sea Foam's web site says you can run the engine on 100% straight Sea Foam, so apparently it's not an issue.
I sure hope new batteries ends up fixing the issue, because a set of batteries for this truck runs about $250. On the other hand, even if it doesn't fix the problem, the batteries are definitely bad (no matter what AutoZone's tester says) if they can go from topped off to totally flat in one day (less than 40 miles of driving total).