I actually wore a hole right through the rubber. What surprised me when I discovered this is that the inner tube was still intact. That's right. The tire wasn't flat. Amazingly good luck.
Since the other tire was worn, but not as badly, I opted to buy two new tires. They arrived yesterday, so I immediately set to getting them mounted onto the utility cart so that I'd be back in business hauling things around the farm again.
I had greased the axels with lithium grease when I mounted the last set of tires, but even so, it took some persuasion to get the old tires off. A hammer and a block of wood came in handy. Once off, I set about cleaning up the axels. Normally I'd use sandpaper but alas, I had none. Rather than dropping everything and heading for the Ace, I improvised. A rasp and steel wool did a decent job of cleaning the axels.
Before mounting the new tire, I applied lithium grease to the axel. A rubber mallet gently got the tire in place.