Car doesn't always start & doesn't stay running.

Engine doesn't always start. Sometimes it starts right up. Most times, when you turn the key forward you hear the fuel pump come on, but it doesn't stop. You have to keep turning the key on & off until you hear the fuel pump stop, then the car will start. Once started, it will idle for awhile but then it just dies. I put a fuel pressure gauge on it, and when idling the pressure stays between 35-40. If it sits overnight the fuel pressure will bleed off, though when the car dies it still has good fuel pressure. I put a new distributor module and fuel filter on but neither made a difference. I also made sure to fill it up with fresh gas & added fuel line cleaner. The car ran great for the previous owner. He drove it from California to Ohio without any problems. The car then sat for 11 months, and ever since that we've had this problem.