Okay, so on the trip it was still doing some odd things: it would cycle on, run for 30-60 seconds, then shut down again. With the new board, however, it wasn't cycling back on as quickly - it'd go for a much more normal amount of time (10-15 minutes, though I didn't time it) before cycling back on again.
I was at Jackson Center over the week, so I picked up a new thermostat/ECO for it, and dropped them in Thursday morning. After that, I never noticed the water heater doing anything strange during the trip again. It would cycle on, run for a few minutes as needed, then cycle off again for a while.
So I'm not entirely sure whether I could've fixed it with just the thermostat. I could swap the old board in and find out, I guess, but I probably won't.
I'm not even sure the thermostat was bad - the foam covering over the thermostat and ECO was shot, and maybe that was affecting it since it wasn't being insulated from the burner's heat as well.
My suggestion for anyone with a similar problem: start by replacing the thermostat and ECO; it's about $20 and easy to do. I'm not convinced the board wasn't going funky on me, too, though. Still, I'll probably toss the old one in the camper somewhere in case the new board dies on a trip - it's probably better than nothing.
Also, on the slow starts: never had a problem during the trip. It fired up immediately every time. Ahhh, that's nice.