There always needs to be an air space in the hot water tank. However, that should not affect cold water use. There should be a check valve at the tank to keep hot water from back flowing into the cold side.
Some water systems have a small pressure tank on the cold side to stop the pump from frequently turning "on" when only a little water is used. Pump will run longer to fill the pressure tank.
A sticking pressure switch inside the pump may stop it from turning "off" until pressure builds higher than normal. "Start" pressure is often lower than "stop" pressure.
Dutchman Kodiak Cub KD176RD 2018
Nissan Pathfinder 2015