Thanks, Jim;
Attached is a photo, though maybe not from the best angle. Note that in the first picture, it was the opposite side tire that had fallen off. By the time we got to the 2nd picture, both tires had fallen off. In the 2nd picture, you can see the tire looks a little "flat". If you zoom in, you might see that the bottom part of the tire is compressed against the rim, while the top part has raised up from the rim. Eventually we disassembled the boat and carried the amas separately. The tires still fell off, but not as frequently. Since posting this question, I discovered the ***'s in Hobie's catalog, and they do not seem to recommend ANY of the Hobie Carts for the island; only the Hobie Dolly...
I think moving forward, I would like to purchase one of the carts I linked above. It's a difficult decision, as the one with the balloon tires is pretty heavy and big, and I'm worried it might have a lot of rolling resistance. For future adventures, we would like to have the ability to take the boat on land, sometimes for long distances.