A couple of updates...
It took a couple of days, but I finally spoke to a tech support person from ASA. They allowed as how the 12 volt adapter was cheap and prone to failure, so they're sending me a new (and more robust, I hope) adapter.
In the meantime, given that ASA thought is was just the adapter, and Pod2Vibe's experience replacing it, I picked up a heavier duty adapter and spliced it into the wire to the TV-end plug. With that done, I'm happily watching the Red Sox - Yankees game. Well, I'd be happier if the Red Sox were winning.
PS, while sorting this out I tried, a couple of times, to watch the little Furrion tv in the bedroom. As near as I can tell, that tv isn't worth the plastic it's made from. Seems like you have to be in precisely the right position, relative to the screen, or else large sections fade to black. I couldn't find one of those positions that didn't involve standing up. Oh well. Maybe I'll get something more usable some day.