As ok4450 mentioned, a new fuel filter would be an immediate first step, and if the fuel pressure is too low then the fuel pressure is too low. Someone should be watching what the fuel pressure does on a hot shut down and
How old are the plugs and wires?
The reason that the thermostat and coolant temp sensor are important is that the car has to run rich (higher than normal fuel in the mix) when it is cold. If the car never gets up to full temp or if the computer doesn’t know that the car is up to temp it will keep the mix rich. In other words, the coolant temp & temp sensor are crucial for fuel mix.