Suddenly, my toilet is making an awful moaning noise when it's flushed. It is mitigated by opening the faucets in the bathroom sinks. I've tried to open all the upstairs faucets in the bathrooms and showers and then flushed the toilet to see if I could get the air out of the system. Didn't solve the problem. How do I get rid of this?

Your toilet ballcock washer's vibrating. This can be caused by trash in the supply line or the washer seat. Pull the tank lid and check the water leve when the tank's fulll. It should be 1/2 inch below the white overflow tube. If the water level's correct then leave the tank top off and watch the float ball while someone turn on a cold water faucet and then shuts it off real fast. Did the float ball bob up and down and the ballcock make a noise? Shut the water off to the toilet and remove the three screws on top of the ballcock and pull up the float and washer stem assembly. Hold your hand over the ballcock and turn the water back on to flush out the supply. M Make sure the washer's seated in its holder and reasw

This isn't air inn the line. The washer's vibrating against the seat.
Pull the tank lid and check the water leve when the tank's fulll. It should be 1/2 inch below the white overflow tube. If the water level's correct then leave the tank top off and watch the float ball while someone turn on a cold water faucet and then shuts it off real fast. Did the float ball bob up and down and the ballcock make a noise?
If the adjustments are OK then shut the water off to the toilet and remove the three screws from the top of the ballcock and pull the float ball,stem and washer assembly up and out. Seat the washer firmly in its holder. Now hold your hand over the open top and turn the water back on for a minute to flush out the supply. Reassemble and test. Better now? Regards, Tom