1995 Chevy Corsica having problems starting back up after r

Our 95 Corsica that we just purchased from individual is having problems with starting back up after running for a short while. You can start it up in the mornings or after it has cooled but then after driving for about 30 minuets and you shut the engine off wait for a min and try to start it back up, it just turns over and over. Now I have figured if you push the accelerator to the floor and start it then it will start but a choky start puttering and putting a small amount of black smoke out smelling fuel during this time. Then it idles out and runs fine and does not shut off until you turn it off and it all starts over same routine. Also the check engine light turns on and off while driving but performs fine. We have put it on a computer and comes up with nothing had multiple people look at it and nothing changes fuel pressure regulator still problem persists fuel pump is good and pressure seems fine. Do not know what to do and this is a major problem for us. Could you please advise us on this? Thank you, Chad