Car goes into 'limp mode' soon after starting?

Car Check Engine Light (CEL) came on and car went into limp mode a few week ago. I checked the car: fluid levels normal, hose clamps tight, electrical connections tight. However, a single wire (tan or white with red stripe) with a plastic female connector was hanging between the firewall and the engine in the engine compartment. Mechanic and Ford service folks could not identify it. (See attached picture) After sitting overnight, the car started normally and ran fine for about 5 minutes, then CEL and limp mode again. I took it to an independent garage and they were unable to pull any error codes from the PCM (powertrain control module) using an OBD1 tester. The Ford dealership could not pull any codes either. I bought an Inova OBD1 reader and tried pulling codes. With key on engine off (koeo) I got a 111 code (all relays working?) And 10 (continue with next test?). With key on engine running (koer) the OBD1 produced no codes and the PCM appeared to lockup (radiator fan came on and ran continuously, CEL plus Battery light and Oil light on Dash). Had to disconnect battery to unlock. Incidence of limp mode is happening sooner and sooner as time goes on. Fuel line pressure seems normal 40 PSI. I'm waiting for delivery of a refurbished PCM, but hope is fading. Help! Suggestions appreciated...

Thanks for responding Ken!

Here are a couple more pictures of the mystery part. In the tilted image, the brass female connector may partially be seen at the center of the connector tip, kind of like a half-moon image. The single wire attached to the plastic connector appears to be either tan colored with a red stripe or dirty-white colored with a red stripe. The Ford technician who looked at the mystery part thought it was 'part of the wiring harness.'

Regarding 'limp mode:' Initially, when the problem first started, the car would drive normally -- very responsive, good power/acceleration when gas pedal depressed, good power going up hills, etc. -- Until the CEL (check engine light) came on, at which point the car immediately became very non-responsive, i.E. Would not respond to depressed gas pedal; was extreeeeemley slow to accelerate; slowed waaay down ( 10mph) when confronted with an uphill situation; would not go above 40mph on a level road. The exhaust also seems to change; it becomes very stinky, possible from very rich fuel mix? I'm no expert on that but it is way more stinky when the car is in limp-mode than it is under normal operating conditions. If the car is allowed to sit for several hours with the engine off, when restarted it behaves normally for 2-5 minutes after which time the CEL comes on and car exhibits 'limp-mode' non-responsiveness again. As the days go by since the problem first occurred, the CEL/limp-mode situation occurs after shorter and shorter engine-running intervals. Regarding the fuel pump; I've measured the fuel line pressure at the engine through the Schrader valve and the PSI is steady right around 40 PSI. The pressure remains steady even when the car is turned off, so I don't think there is a leak in the fuel line.

Ken, can you tell me what you're looking for in a video? I will be happy to post one for you. Again, many thanks!

Hi Ken, I didn’t put the car on jacks but I did reach under and bang on the exhaust system and catalytic converters with a rubber mallet. I did not hear anything rattling around. Also, wouldn’t the exhaust smell like rotten eggs (sulfur) if the cats were bad? The exhaust is stinky but not like sulfur.

I’ve added some pictures of the "mystery part" for you to look at. I hope they help.

I’ve got a semi-related question for you. When a PCM puts a car into "limp mode, " what is it doing to cause the reduction of power in the engine? Is it somehow limiting the fuel and/or the gas? I can’t find the answer to this question anywhere on the internet.

Thanks again for all your help Ken!

Hi Ken, The problem turned out to be a faulty Powertrain Control Module (PCM). I guess the first clue was that no one could pull any error codes with an OBD1 reader. I received a refurbished and pre-programed PCM from FlagShip-1 today and installed it in the car. The car started right up and drove great. I put it through its paces and had not hint of sluggishness or surging. It was very responsive to acceleration, etc. And no sign of that danged CEL! Thanks again for all your help with my 1995 Ford Taurus station wagon!