Engine cuts off

Start at checking the fuel pressure if its within specs-fuel pump could be failing

The coolant temperature sensor may be bad. If you have a temperature gauge, there are genrally 2 coolant sensors. One for the gauge and one for the coolant temperature signal to the ECU. The sensor to the ECU sensor may be stuck in a position where it thiks the coolant is always belwo operatni temperature. This will affect the fuel curve and the fuel pressure may not be adequate to keep the car running correctly.
The other possibilities are the Idle Air control Valve, IAC and the Fast Idle Thermo Valve, FIT.
The FIT acts as a choke and in when starting a car it leans out the air fuel ratio. It may be stuck in this position which is fine for startup, but when the engine is warm it is like a vacuum leak and will affect performance and cause the car to run rough or stall. The Idle Air Control Valve can be integrated with the FIT function built into it. It keeps the idle at a steady state and chnges the throttle position and/or the A/F ratio by changin passages in the throttle body that let air in from in front of the butterfly of the throttle body.
It can be electrically controlled or vacuum controlled. Sometimes the vacuum line can get a leak and cause idle and stalling issues as it does not have enoguh vacuum to keep the idle up or the passage opening at the poper set point to keep motor running correctly. If the IAC has an elctrical connector you can test it by removing it and turning the key to the, "ON" position and seeing if it moves. It can also become dirty with carbon deposits overtime and need cleaning with carberator cleaner and a rag. When cleaning passages in the throttle body or intake manifold, spray cleaner from outside in towards the intake direction to prevent deposits from going deeper into passages and causing furhter clogging. It may also have coolant lines running through it and if so equipped the FIT will be next to it and have coolant lines running through it as well. Thisis to change with changes in coolant temperature.
Also you may want to check fuel pressure and have the trouble codes pulled as a check engine light does not have to be on for a code to be present or stored in the ECU. Advance Auto or Auto Zone will pull codes for free.