137 is Oxygen sensor not switching/system rich Left or Front HO2S - Fuel control
The other code 538 popped up because you didn't press the gas pedal during the test (ignore that one for now and repeat the test properly later).
WOT makes it run better may mean that a sensor is bad. WOT puts the computer back into open loop, by design. PCM ignores sensor inputs and runs off of pre-programmed values.
The easy things that need tested are the fuel pressure regulator
http://www.fordfuelinjection.com/?p=41
However, that link does not really address the vacuum control aspect of the FPR. Once you have the gage hooked up and reading pressure, run the engine at idle, note the fuel pressure. Goose the gas to a pretty good rpm 3k or so. Note that the fuel pressure increases (as the manifold vacuum decreases) with engine rpm increase. You can also pull the vacuum line off of the FPR (engine running, plug the line with your finger) and see the same type of pressure increase plus it should run rougher (like yours already does). See if that removed vacuum line has gasoline in it (engine off) meaning FPR is leaking and bad.
the IAT (intake air temp sensor)
http://www.fordfuelinjection.com/?p=29
the engine coolant temp sensor
http://www.fordfuelinjection.com/?p=28
Don't confuse the two coolant temp sensors, you want to test the 2 wire one that feeds the PCM (not the one-wire sensor for the gage).
I would do the tests in the order that I listed them.
Very few things will make it run rich yet run OK at WOT. I'm betting that one of those three items is gonna test bad.