Thanks, all. Yes, that link in the first reply is where I began this morning. I did the login and worked through a succession of screens: SLOWLY filled in all the countries for the one phone (really, really awful interface!) ... and then the next screen switched me to (only) the $10 day plan.
We'll be gone 60+ days. If I could trust that it would really be a "per day" deal then, sure, it might work out about the same as a couple of months on the cheapo plan. But I've read some horror stories (one I think was here on RS) from people who THOUGHT they'd only used a couple of days, only to find that AT&T "counted" a weak WiFi connection as a usage day. I know, I know ... sounds fishy and there are always horror stories.
In the past we've bought one plan for two months to cover "emergencies" -- and never had to use much of the data allocation at all. We plan to stay in Airplane Mode on both phones 99% of the time. But we're staying in AirBnB-type places and need to make a connection to the host on arrival in each city. I'm hoping texting will work for that, and that no odd calls come in while I have the phone switched on.
And then there's the issue of preventing voice mail from firing up whenever the phone is switched on. But that's another question/topic.
I'll sleep on it and try the phone contact tomorrow.