Is this only cold engine starts?

Before you tear apart the car, check the pressure at the fuel rail. After sitting for at least 4 hours, the pressure should be close to zero. If the pressure is less than 10 lbs a few minutes after engine shutdown, then the check valve is bad. But, the check valve is now part of the fuel pump, and you don't want to go there. Remember, no check valve will hold pressure. It is just there to hold fuel in the line.

Your car also has a PEM, probably external and exposed to the elements by the fuel filter. If so, you need a VIDA scan to diagnose it's codes.

If the PEM has been moved to the inside, there is a dealer service writeup to do that to cars with failing PEMs, then that should be just fine unless it got wet.
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Keeping it running is better than buying new