What could be causing this drop in water pressure?
My guess is there is a restriction in the line going to the house, with only enough getting through to feed one thing, either the toilet or the faucet. And since the toilet is probably drawing the water first before the faucet, it goes there.

I agree that it is more likely just heavy rain that is causing mud in your water. However, surface water getting into your water supply is bad news. You should consider adding filters to clean up the water supplied to the house. I'd guess a chlorine injector to kill any bacteria in the water, a tank to give the chlorine some time to do its job and let mud settle out, coupled with a filter that can take the remaining mud and chlorine out.