Thank you all for the feedback.
I'm going to look into replacing my 27 inch wheels with 700C wheels.
Folding 27" tires are available here and bringing a couple of them is an option I have also been considering. From what you all have posted, that may be the safest option if I can't fit new wheels on the old frame.
My first derailleur bike, 1973 Jeunet, had 27" wheels, as did all the European bikes sold in the USA in that era. I gather that the change to 700C was made to facilitate switch-overs from clincher-rim wheels to tubular-rim wheels. But who uses tubulars to tour in, or for that matter for any sort of general recreational cycling? Sounds like planned obsolescence, but what can one do?
As of this writing, I would not hesitate to take off touring in this country with my bike as is, but I can see that the writing is on the wall for 27" wheels/tires here as well. Maybe another 10 years, and they will be on the obsolete shelf beside my collection of 8-track tapes.
Thanks for the offer to hold a tire for me, gilespargiter! I expect to begin this trip sometime in mid to late April, but haven't fixed an arrival time or airport. If I run into any problems like this, I will PM you as you suggested.