Need help with strange problem on '95 Fairmont

My 1995 Fairmont will intermittently cut out while driving. This usually happens when the engine is at operating temp and after a short stop (like quickly stopping at the shop to buy milk or something). It also normally happens shortly after driving off while the speed is still slow. If I try to restart, the engine will start but it runs very rough (actually sputters) and cut out again. This is when I start it from the On position on the ignition. Only if I turn the key in the ignition all the way to Off, and start it then, will it start normally and run as if nothing is wrong.

I pulled the error codes and it does not report anything wrong, except code 126 which is MAP sensor out of range. I recently replaced the MAP sensor with a new one and I doubt that it could be the problem. The previous MAP sensor was faulty and I had almost no power, the engine was running rich the whole time, and I only got 5 km to the litre (20 litre per 100km) on the open road. Since replacing the MAP sensor, the power is great, and fuel consumption is now 10km to the litre (10 litre per 100km) on the open road.

Before the MAP sensor replacement, the car did not cut out like it does now. This only started after the MAP sensor replacement.

Any ideas what may be the problem?