"Does anyone travel with an old school watch (that we don't own) just to be sure that one time piece is accurate and not subject to changing automatically based on your geographic location and a connection to a tower?"
I do. It's a citizens chronograph. Battery operated, and I have to set the watch back 1 hour, forward 1 hour, etc. And it's never been a problem for me.
One thing to keep in mind. Here in the U.S. we recently went back on Daylight Savings Time. It's an artificial time which I despise. So things get really screwed up, considering many countries and island nations never go off Standard Time. I know Mexico doesn't. So even though I am in the Central Time Zone as Cozumel is, they have an hour difference. For example, as I type this, it's 10:06CDT here in Texas, but in Cozumel, it is 9:06CST......even though we are in the same time zone, and hour difference. Stuff like this drives me nuts. For me, it's NOT a simple hurdle to overcome, when trying to stay on ships time vs. local time when you go on an excursion.
Sorry I don't have an answer for you.... I just needed to vent.