After a hot soak, pull the vacuum line on the fuel pressure damper on the back of the fuel injector manifold. See if any fuel is in the vacuum line..

 

I stop when the nozzle shuts off. And I will check out the vacuum line.

 

The vacuum line did not appear to have gas in it.

 

More information: today under the normal circumstances that the issue occurs, it didn't happen but instead it started up easy then for about 30-45 seconds the engine rpm went between about 700 and 1500 but was not accompanied by the lights flickering then it drove perfectly fine.

 

My daughter's 2004 Escape V-6 is doing the same exact thing. We've had 2 overnight heavy rainstorms about a week apart. Car cranks well but will only start with the gas pedal fully depressed. Ran some injector cleaner through the last 1/2 tank of gas between deluges, but it did the same thing yesterday morning.

I am wondering about cleaning out the throttle body. Need to open it up for access to check for carbon. It is clearly a fuel problem and not electronic as I am getting no trouble codes for any type of misfires.

 

Daughter's car acted up again but in dry weather. Threw a P0403 code, which indicated the EGR valve problem. I started looking over the EGR valve making sure the vac hoses were tight (I'd replaced them all about 6 weeks ago as several were dry rotted). Just happened to reach around the back side of the EGR valve and the nut holding the EGR pickup tube was loose. Loose as in I was able to remove it by hand. Put the tube in all the way and tightened the nut down. Car idled and ran fine.