Just got off the phone with a guy at Thomas Distributing, a major seller of rechargeable batteries. They're pretty knowledgeable about this stuff. I told him what the deal was and he said it doesn't surprise him that a low-drain device would not like NiMH batteries. He said those batteries are designed for high-drain devices and they won't work as well as standard alkaline batteries in low-drain devices.

That makes me feel a little better, but it still doesn't explain why I'm the only one having problems.