I am going to assume that you have a 4.0L injected motor now. You did not specify what engine you put in, so here we go.
This sounds like a sensor problem or a vacuum leak or a relay. I would check all of the vacuum lines and the intake seal to the gasket and head first. Also take a look at the throttle body seal to the intake.
A really easy way to check this is with a can of carburetor cleaner. Start the engine and then start spraying the cleaner around all of the suspect areas. If the motor starts to hesitate or die (not from flooding) you have found the leak. Be very careful when spraying the cleaner towards the exhaust. It will ignite and burn if the exhaust is hot. Some people prefer to use an unlit propane torch. This has the opposite effect. The motor will start to race when you find the leak. I DO NOT recommend using propane, it is just too hazardous.
If you can’t find a leak and everything seems to be tight you will need to get the sensors checked. Best thing to do for this is take it to a shop and have them put it on a diagnostic computer and check it out. Any time that you do an engine swap and the motor is controlled by a computer, I would recommend getting a diagnostic done on it just to be safe even if everything is new.
Another possibility is the regulator on the fuel rail and its relay. At the front of the fuel rail there is a valve that will close once the rail is up to pressure. You should be able to hear your fuel pump kick on when you turn the key to the run position (don’t start it) and then it should stop. If this is not happening try swapping the relay with another one and see if it shuts the pump off. If the pump is staying after you put a known good relay the valve/regulator on the fuel rail may be bad. There is a way to test the pressure on the fuel rail but I am not sure how to do it. I think XJNick may be able to help you there.