We have good news and bad news for you, Eric. The good news, according to University of Minnesota turf expert Bob Mugaas, is that while mushrooms can look unsightly (and they're worse in years with a lot of rain), they're actually beneficial to your lawn. “They're part of the breakdown of organic material in the soil, and they help recycle nutrients.”
The bad news, as you've already discovered, is that mushrooms are nearly impossible to get rid of. They're actually the fruit of an extensive underground root system. So even if you remove the visible mushrooms or use fungicides, the source of them is still there (they're like the tip of an iceberg).
According to Mugaas, you have several options. “You can certainly pull them.” This won't permanently rid your lawn of mushrooms, but it can give you temporary relief.
You can also make your lawn less hospitable to fungi by correcting drainage problems and eliminating decaying organic matter. Grind down stumps, rake up grass clippings, dig up buried lumber, aerate, dethatch and replace old mulch.
The easiest option (or maybe the hardest for you, Eric) is to make peace with your mushrooms. Their numbers will increase and decrease depending on the season. Teach your grandchildren never to eat mushrooms from the lawn, and during cool, wet periods, keep a close eye on your pets.