Looking at the symptoms: pathetic water flow, and air. All faucets and the showerhead affected. Same problem whether on city water or water pump.
I'd look behind the basement wall. Take the wall down behind the UDC and trace the output of the water pump to where it tees into the city water line. Then find where the line splits out between kitchen and bathroom. Somewhere between those 2 points is the area in common to all these symptoms. Air coming out the faucet suggests an air source/leak in that area. Have DW run the water and listen for air sounds. Look for leaks.
Have you noticed any issues with hot water? Turn off the water heater and throw the levers to bypass it. I wouldn't think it's the problem, but there is a plastic check valve on the hot water outlet (top) that sometimes cracks, causing leaks. Usually it leaks water, but maybe it can fail in such a way as to allow air in the line.