Forgive me for taking so long to reply. Your and Carmel's Questions, with my Answers:
Q. Does this always happen when changing zoom levels?
A. No, it only seems to be happening when I first open the map. Once I zoom or pan, the problem seems to be fixed and doesn't recur until I refresh the map, or click the Modify Map option in the upper left. And it doesn't always happen when you first open the map. (Based on only a half dozen attempts).
Q. Is layer optimization enabled?
A. Yes. Setting that didn't seem to change anything.
Q. How many vertices are there with each layer.
A. There are only two layers. The larger one has about 250,000 vertices. The smaller one maybe another 50,000. I think that counts as smallish, no?
Q. Do you notice any issues if you try in other browsers (Internet Explorer, Firefox, Google Chrome). I'm running Chrome. I tried opening the map from an incognito window on a Chromebook, and the problem did happen. I successfully opened the map on an Apple without the problem, but with only one attempt, so who knows.
Q. Instead of signing in, does the issue appear if you choose Modify Map:
A. Yes. I was in incognito mode in a Chrome browser and opened the map. It happened to open up without any doubling problem, but went double when I clicked on Modify map.
Thank you both for having looked at this, and Richard for reassuring me that it wasn't all in my head!