If the pump is pumping water, but there is air also at the faucets, then the air has to be entering the system BEFORE the water pump. The line between the fresh water tank and the water pump is broken or cracked, causing the pump to suck in air. The pump will not shut off if it's sucking in air.
If the line is broken or cracked AFTER the pump, then you'll have evidence of a leak. You won't get air at the faucet, it will just be a slow stream. Air will not enter when water is being pushed out.
The air intake is occurring between the fresh water tank and the pump.