I've had the same problem, but I just don't try to grow anything else under the feeders, and let some of the seedlings of the sunflowers grow when they pop up beneath the feeders. I would suggest planting dwarf sunflowers under your feeders, but it sounds like yours are in a shady area. The only thing I can recommend is to buy sunflower chips, or no-waste seed, which is more expensive, but is hulless. Don't stop feeding the birds sunflower seeds, though, or you'll lose all the prettiest birds! Sunflower is the food that attracts the most varieties of birds, although nyger will attract finches and safflower is a favorite of cardinals. Doves like millet, but so do pigeons. And corn will just get you more squirrels!

Perhaps if you start adding compost to the area around the river rock, it will neutralize the problem. That's what I did to the area where I formerly had my feeders. I had to move them because they were too near the house and mice were stealing the seed and carrying it into our garage! It's been a few years now, and all the plants have filled in. I also noticed that the worms really liked the area with the mixed compost and hulls, maybe because it was easy to work through. They probably helped get the soil in better shape, too.