New fuel pump but tries to shut off at idle

Recently had fuel pump changed. Check engine light still on. Check gas cap light came on while starting home. It went out a couple days later, but is now back on. Sometimes car idles fine, but more often not. Sometimes hard to re-start. Sometimes shuts off at stop. Bad pump maybe? Even though it is new, the mechanic sent the first one back, not enough fuel pressure. I don't know the odds of getting two in a row bad. Bad seal at fuel pump? The car is tending to run better with full gas tank, gets worse as gas is used up and it gets under 1/2 tank.