The settling is fine and normal. You have to rinse off eggs that come out of the container before use anyway, so don't worry about any on the eggs themselves. The purpose of the lime is to make the water very alkaline and protect against bacteria access to the eggs. Fortunately, the lime is cheap.
Are you using the mix: 1oz (weight) of lime / quart of water?
Theory:
I suspect that the standard mix is just too high on the lime which simply results in the excess not dissolving. Not harmful, but wasteful. I have not tested the ideal mix to minimize settling, what effect is has on PH and how it might affect egg preservation.